Proof:. Integrating 2 we find that since \(\Phi(0)=0\), \[\begin{align} C_{\Phi}\Phi(s)= C_\Phi\big(\Phi(s)-\Phi(0)\big)=C_\Phi\int_0^s \Phi'(\sigma)\mathop{\mathrm{d\!}}\sigma\geq\int_0^s \sigma \Phi''(\sigma)\mathop{\mathrm{d\!}}\sigma\quad\text{for }s\in[0,s_0]. \end{align}\] In view of an integration by parts on the right, again using \(\Phi(0)=0\), entails \[\begin{align} C_\Phi\Phi(s)\geq s\Phi'(s)-\int_0^s \Phi'(\sigma)\mathop{\mathrm{d\!}}\sigma= s\Phi'(s)-\Phi(s) \end{align}\] for \(s\in[0,s_0]\). Therefore, we conclude that \[\begin{align} s\Phi'(s)\leq (C_\Phi+1)\Phi(s)=:C_1\Phi(s)\quad\text{for all }s\in[0,s_0] \end{align}\] and hence obtain 14 . As for 15 , we note that the claim is obviously true for \(s=0\), while for \(s>0\) 2 is equivalent to \[\begin{align} \big(\ln\big(\Phi'(s)\big)\big)'\leq \big(\ln\big(s^{C_\Phi}\big)\big)'\quad\text{for all }s\in(0,s_0]. \end{align}\] Integrating this from \(s\) to \(\alpha s\) for \(s\in\big(0,\frac{s_0}{\alpha}\big]\) we find that \[\begin{align} \ln\big(\Phi'(\alpha s)\big)-\ln\big(\Phi'(s)\big)\leq \ln\big((\alpha s)^{C_\Phi}\big)-\ln\big(s^{C_\Phi}\big)\quad\text{for all }s\in\big(0,\frac{s_0}{\alpha}\big], \end{align}\] which entails \[\frac{\Phi'(\alpha s)}{\Phi'(s)}\leq\alpha^{C_\Phi}=:C_2\quad\text{for all }s\in\big(0,\frac{s_0}{\alpha}\big].\] Finally, assuming \(s_1>s_0\), the extension of the inequalities 14 and 15 for \(s\in[\frac{s_0}{\alpha},s_1]\) and \(s\in[s_0,s_1]\), respectively, is then an evident consequence of the fact that by the convexity of \(\Phi\) the number \(C_{3}=C_{3}(\Phi,s_1):=\max_{s\in[s_0,s_1]}\Phi''(s)\geq 0\) is well-defined, and the fact that \(\Phi'\) and \(\Phi\) are both monotonically increasing. Taking \(C=C(\Phi,\alpha,s_1):=\max\big\{C_1,C_2,\frac{\Phi'(\alpha s_1)}{\Phi'(\tfrac{s_0}{\alpha})},\frac{C_3s_1}{\Phi'(s_0)},\frac{\Phi'(s_1)s_1}{\Phi(s_0)}\big\}\) we arrive at the asserted inequalities in 16 . ◻

Proof:. Denoting by \(\bar{\varphi}:=\frac{1}{|\Omega|}\int_{\Omega}\!\varphi\) the spatial average of \(\varphi\), we make use of Hölder’s inequality to estimate \[\begin{align} \int_0^{T}\!\!\int_{\Omega}\!|\varphi|^p=\iint\limits_{\{|\varphi|>0\}}|\varphi|^p&\leq\Big(\iint\limits_{\{|\varphi|>0\}}|\varphi|^\frac{p(p+N)}{N}\Big)^\frac{N}{p+N}\Big(\iint\limits_{\{|\varphi|>0\}}1\Big)^\frac{p}{p+N}\\ &\leq 2^p\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\Big(\int_0^{T}\!\!\int_{\Omega}\!|\varphi-\bar{\varphi}|^\frac{p(p+N)}{N}+\int_0^{T}\!\!\int_{\Omega}\!|\bar{\varphi}|^\frac{p(p+N)}{N}\Big)^\frac{N}{p+N}. \end{align}\] Drawing on the Gagliardo–Nirenberg inequalitywe find \(C_1=C_1(\Omega,N,p)>0\) such that for all \(\varphi\) of said regularity class, \[\begin{align} \int_0^{T}\!\!\int_{\Omega}\!|\varphi|^p\leq 2^p\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\Big(C_1\int_0^{T}\!\big\|\varphi-\bar{\varphi}\big\|_{W^{1,p}(\Omega)}^p\big\|\varphi-\bar{\varphi}\|_{L^{p}(\Omega)}^\frac{p^2}{N}+\int_0^{T}\!\big\|\bar{\varphi}\big\|_{L^{\frac{p(p+N)}{N}}(\Omega)}^\frac{p(p+N)}{N}\Big)^\frac{N}{p+N}. \end{align}\] Herein, we can rely on a Poincaré–Wirtinger inequality to obtain a constant \(C_2=C_2(N,\Omega,p)>0\) such that \(\|\varphi-\bar{\varphi}\|_{W^{1,p}(\Omega)}^p\leq C_2\|\nabla\varphi\|_{L^{p}(\Omega)}^p\), so that with \(C_3=C_3(N,\Omega,p):=2^p(1+C_1C_2)^\frac{N}{p+N}>0\) \[\begin{align} \label{eq:emb-eq-1} \int_0^{T}\!\!\int_{\Omega}\!|\varphi|^p&\leq C_3\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\Big(\int_0^{T}\!\big\|\nabla\varphi\big\|_{L^{p}(\Omega)}^p\big\|\varphi-\bar{\varphi}\|_{L^{p}(\Omega)}^\frac{p^2}{N}+\int_0^{T}\!\big\|\bar{\varphi}\big\|_{L^{\frac{p(p+N)}{N}}(\Omega)}^\frac{p(p+N)}{N}\Big)^\frac{N}{p+N}\nonumber\\ &=:C_3\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\big(I_1+I_2\big)^\frac{N}{p+N}. \end{align}\tag{17}\] To estimate further, we note that for arbitrary \(q\geq1\) \[\begin{align} \label{eq:est-barphi-eq1} \|\bar{\varphi}\|_{L^{q}(\Omega)}\leq |\Omega|^{\frac{1}{q}-\frac{1}{p}}\|\varphi\|_{L^{p}(\Omega)}\quad\text{ and }\quad \big\|\varphi-\bar{\varphi}\big\|_{L^{p}(\Omega)}\leq 2\|\varphi\|_{L^{p}(\Omega)}. \end{align}\tag{18}\] Accordingly, using the assumption \(T\leq 1\) we find that \[\begin{align} I_2\leq\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\|_{L^{p}(\Omega)}^\frac{p(p+N)}{N}|\Omega|^{-\frac{p}{N}}, \end{align}\] whereas for \(I_1\) we draw on Young’s inequality in addition to 18 and conclude that there is \(C_4=C_4(N,\Omega,p)>0\) such that \[\begin{align} I_1&\leq 2^\frac{p^2}{N}\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^\frac{p^2}{N}\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p\leq C_4\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^\frac{p(p+N)}{N}+\Big(\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p\Big)^{\frac{p+N}{N}}. \end{align}\] Plugging these estimates back into 17 and rearranging exponents we obtain \(C_5=C_5(N,\Omega,p)>0\) satisfying \[\begin{align} \int_0^{T}\!\!\int_{\Omega}\!|\varphi|^p&\leq C_5\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\Big(\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^\frac{p(p+N)}{N}+\Big(\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p\Big)^{\frac{p+N}{N}}\Big)^{\frac{N}{p+N}\cdot\frac{p}{p}}\\ &\leq C_5\big|\big\{|\varphi|>0\big\}\big|^\frac{p}{p+N}\Big(\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\|_{L^{p}(\Omega)}+\Big(\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p\Big)^\frac{1}{p}\Big)^{p} \end{align}\] for any \(\varphi\in L^{\infty\;\!}\!\left((0,T);L^{p}(\Omega)\right)\cap L^{p\;\!}\!\big((0,T);W^{1,p}(\Omega)\big)\). ◻

Proof:. Similarly to the steps in the corollary above, we make use of the Minkowski, Gagliardo–Nirenberg and Poincaré–Wirtinger inequalities, to first find \(C_1=C_1(N,\Omega,p,q_1,q_2)>0\) such that \[\begin{align} \big\|\varphi\big\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2}(\Omega)\right)}&\leq2^{\frac{q_1-1}{q_1}}\Big(\int_0^{T}\!\|\varphi-\bar{\varphi}\|_{L^{q_2}(\Omega)}^{q_1}+\int_0^{T}\!\|\bar{\varphi}\|_{L^{q_2}(\Omega)}^{q_1}\Big)^\frac{1}{q_1}\\ &\leq C_1\Big(\int_0^{T}\!\|\nabla\varphi\|_{L^{p}(\Omega)}^{q_1\alpha}\|\varphi-\bar{\varphi}\|_{L^{p}(\Omega)}^{q_1(1-\alpha)}+\int_0^{T}\!\big\|\bar{\varphi}\big\|_{L^{q_2}(\Omega)}^{q_1}\Big)^{\frac{1}{q_1}}, \end{align}\] with \[\alpha=\frac{N}{p}-\frac{N}{q_2}=\frac{p}{q_1}\quad\text{and}\quad 1-\alpha=1-\frac{p}{q_1}.\] Here, the assumption \(T\leq 1\) and the estimates \[\begin{align} \|\bar{\varphi}\|_{L^{q}(\Omega)}\leq |\Omega|^{\frac{1}{q}-\frac{1}{p}}\|\varphi\|_{L^{p}(\Omega)},\;q\geq1, \quad\text{and}\quad \big\|\varphi-\bar{\varphi}\big\|_{L^{p}(\Omega)}\leq 2\|\varphi\|_{L^{p}(\Omega)}, \end{align}\] entail that there is \(C_2=C_2(N,\Omega,p,q_1,q_2)>0\) such that \[\begin{align} \big\|\varphi\big\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2}(\Omega)\right)}&\leq C_2\Big(\int_0^{T}\!\|\nabla\varphi\|_{L^{p}(\Omega)}^p\|\varphi\|_{L^{p}(\Omega)}^{q_1-p}+\int_0^{T}\!\|\varphi\|_{L^{p}(\Omega)}^{q_1}\Big)^\frac{1}{q_1}\\ &\leq C_2\Big(\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^{q_1-p}\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p+\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^{q_1}\Big)^\frac{1}{q_1}. \end{align}\] Making use of Young’s inequality with exponents \(\frac{q_1}{p},\frac{q_1}{q_1-p}>1\) yields \(C_3=C_3(N,\Omega,p,q_1,q_2)>0\) satisfying \[\begin{align} \big\|\varphi\big\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2}(\Omega)\right)}&\leq C_3\Big(\mathop{\mathrm{ess\, sup}}_{t\in(0,T)}\big\|\varphi(\cdot,t)\big\|_{L^{p}(\Omega)}^{q_1}+\Big(\int_0^{T}\!\!\int_{\Omega}\!\big|\nabla\varphi\big|^p\Big)^\frac{q_1}{p}\Big)^\frac{1}{q_1}, \end{align}\] from which we conclude the claim upon rearranging exponents. ◻

Proof:. We pick any cut-off function \(\psi\in C_0^\infty(\Omega_T)\) satisfying \(0\leq \psi\leq 1\) in \(\Omega_T\) and \(\psi\equiv0\) in \(\Omega_T\setminus \overline{Q_r}\) with \(\psi_{\vert\partial_p Q_r}=0\) and set \(\Psi_h:=\pm\big(\Phi\big([\Phi^{-1}(w)]_h\big)-K\big)_\pm\psi^2\). Notice that by the assumptions on \(\Phi\), \(\Phi^{-1}\) and \(w\) we may use \(\Psi_h\) as test function in 7 . Integrating from \(T_0\) to \(T_1\) and recalling \(\Lambda_h^\pm\) from 22 we find that, akin to the reasoning undertaken in [22], writing \[\pm\iint\limits_{Q_r}\Big(\big[\Phi^{-1}(w)\big]_h\Big)_t\Big(\Phi\big(\big[\Phi^{-1}(w)\big]_h\big)-K\Big)_\pm\psi^2=\iint\limits_{Q_r}\big(\Lambda_h^\pm(w)\big)_t\psi^2,\] integrating by parts with respect to time and taking the limit \(h\to0\), we find that that with \(\Lambda^\pm\) given by 21 we have \[\begin{align} \label{eq:loc-en-43-est1} 0&=\int_{\mathcal{K}_r}\Lambda^\pm\big(w(\cdot,T_1)\big)\psi^2(\cdot,T_1)-\iint\limits_{Q_{r}}\Lambda^\pm(w)\big(\psi^2\big)_t\pm\iint\limits_{Q_{r}}\nabla w\cdot\nabla\big((w-K)_\pm\psi^2)\nonumber\\ &\pm\iint\limits_{Q_{r}}a\Phi^{-1}(w)\cdot\nabla\big((w-K)_\pm\psi^2\big)\mp\iint\limits_{Q_{r}}b\big(\cdot,\cdot,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\nonumber\\ &=:I^\pm_1+ I^\pm_2+ I^\pm_3+ I^\pm_4+I^\pm_5. \end{align}\tag{24}\] Making use of \(\nabla w\cdot\nabla(w-K)_\pm=\pm\big|\nabla(w-K)_\pm\big|^2\) and \((w-K)_\pm\nabla w=\pm(w-K)_\pm\nabla(w-K)_\pm\), we can rewrite \(I^\pm_3\) as \[\begin{align} \label{eq:loc-en-43-est3} I^\pm_3&=\pm\iint\limits_{Q_{r}}\nabla w\cdot\nabla(w-K)_\pm\psi^2 \pm 2\iint\limits_{Q_{r}}(w-K)_\pm\psi\big(\nabla w\cdot\nabla\psi\big)\nonumber\\ &=\iint\limits_{Q_{r}}\Big(\nabla\big((w-K)_\pm\psi\big)-(w-K)_\pm\nabla\psi\Big)\Big(\nabla\big((w-K)_\pm\psi\big)+(w-K)_\pm\nabla\psi\Big)\nonumber\\ &=\iint\limits_{Q_{r}}\big|\nabla\big((w-K)_\pm\psi\big)\big|^2 - \iint\limits_{Q_{r}}(w-K)_\pm^2|\nabla\psi|^2. \end{align}\tag{25}\] For the integral \(I^\pm_4\) in 24 we draw on Young’s inequality so that writing \(A^\pm_{K,r}:=A^\pm_{K,r,x_0,T_0,T_1}\) as in 20 , we have \[\begin{align} \label{eq:loc-en-43-est4} |I^\pm_4|&=\Big\vert\iint\limits_{Q_{r}}a\Phi^{-1}(w)\cdot \nabla\big((w-K)_\pm\psi^2\big)\Big\vert\nonumber\\ &\leq\iint\limits_{Q_{r}}\big\vert a\Phi^{-1}(w)\nabla\big((w-K)_\pm\psi\big)\psi\big\vert+\iint\limits_{Q_{r}}\big\vert a\Phi^{-1}(w)(w-K)_\pm\psi\nabla\psi\big\vert\nonumber\\ &\leq\frac{1}{2}\iint\limits_{Q_{r}}\big|\nabla\big((w-K)_\pm\psi\big)\big|^2+\frac{1}{2}\iint\limits_{Q_{r}}(w-K)_\pm^2|\nabla\psi|^2+\iint\limits_{A^\pm_{K,r}}|a|^2\big|\Phi^{-1}(w)\big|^2\psi^2\nonumber\\ &\leq\frac{1}{2}\iint\limits_{Q_{r}}\big|\nabla\big((w-K)_\pm\psi\big)\big|^2+\frac{1}{2}\iint\limits_{Q_{r}}(w-K)_\pm^2|\nabla\psi|^2+\big(\Phi^{-1}(M)\big)^2\iint\limits_{A^\pm_{K,r}}|a|^2, \end{align}\tag{26}\] due to \(\Phi^{-1}\) being monotonically increasing, \(0\leq\psi\leq1\) and \(0\leq w\leq M\). Moreover, we conclude from \(0\leq\psi\leq 1\), \((w-K)_\pm\leq 2M\) and \(|b(x,t,\xi)|\leq \hat{b}(x,t)\) for a.e. \((x,t,\xi)\in\Omega_T\times\mathbb{R}\), that \(I^\pm_5\) satisfies \[\begin{align} \label{eq:loc-en-43-est5} |I^\pm_5|=\Big\vert\iint\limits_{Q_{r}}b\big(\cdot,\cdot,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\Big\vert\leq 2M\iint\limits_{A^\pm_{K,r}}|\hat{b}|. \end{align}\tag{27}\] Therefore, collecting 24 –27 shows that \[\begin{align} &\int_{\mathcal{K}_r}\Lambda^\pm\big(w(\cdot,T_1)\big)\psi^2(\cdot,T_1)+\frac{1}{2}\iint\limits_{Q_{r}}\big|\nabla\big((w-K)_\pm\psi\big)\big|^2\\ \leq\;&\!\iint\limits_{Q_{r}}\Lambda^\pm(w)\big(\psi^2\big)_t+\frac{3}{2}\!\iint\limits_{Q_{r}}(w-K)_\pm^2|\nabla\psi|^2+\big(\Phi^{-1}(M)\big)^2\!\iint\limits_{A^\pm_{K,r}}|a|^2+2M\iint\limits_{A^\pm_{K,r}}|\hat{b}|.\nonumber \end{align}\] To further estimate the last two integrals on the right hand side, we make use of two applications of Hölder’s inequality to see that for \(q_1,q_2>1\) we have \[\begin{align} \iint\limits_{A^\pm_{K,r}}|a|^2&\leq \int_{T_0}^{T_1}\!\Big(\int_{A^\pm_{K,r}(t)}\!\!|a(\cdot,t)|^{2q_2}\Big)^\frac{1}{q_2}\Big(\int_{A^\pm_{K,r}(t)}\!\! 1\Big)^{\frac{q_2-1}{q_2}}\\ &\leq\int_{T_0}^{T_1}\|a(\cdot,t)\|_{L^{2q_2\;\!}\!\left(\mathcal{K}_r\right)}^2\big|A_{K,r}^\pm(t)\big|^{\frac{q_2-1}{q_2}}\\ &\leq \Big(\int_{T_0}^{T_1}\!\|a(\cdot,t)\|_{L^{2q_2\;\!}\!\left(\mathcal{K}_r\right)}^{2q_1}\Big)^\frac{1}{q_1}\Big(\int_{T_0}^{T_1}\!\big|A_{K,r}^\pm(t)\big|^{\frac{q_1(q_2-1)}{q_2(q_1-1)}}\Big)^\frac{q_1-1}{q_1}\\ &\leq\|a\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2\;\!}\!(\Omega;\mathbb{R}^n)\right)}^2\Big(\int_{T_0}^{T_1}\!\big|A^\pm_{K,r}(t)\big|^{\frac{q_1(q_2-1)}{q_2(q_1-1)}}\Big)^\frac{q_1-1}{q_1} \end{align}\] and \[\begin{align} \iint\limits_{A^\pm_{K,r}}|\hat{b}|\leq\|\hat{b}\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2\;\!}\!\left(\Omega\right)\right)}\Big(\int_{T_0}^{T_1}\!\big|A^\pm_{K,r}(t)\big|^{\frac{q_1(q_2-1)}{q_2(q_1-1)}}\Big)^\frac{q_1-1}{q_1}, \end{align}\] which proves 23 with constant \(C=C(\Phi,M,q_1,q_2,a,\hat{b})>0\) explicitly given by \[C:=\big(\Phi^{-1}(M)\big)^2\|a\|^2_{L^{2q_1\;\!}\!((0,T);L^{2q_2\;\!}\!\left(\Omega;\mathbb{R}^N\right))}+2M\|\hat{b}\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2\;\!}\!\left(\Omega\right)\right)}. \qedhere\] ◻

Proof:. Again, first arguing formally using Steklov averages, i.e. taking \[\Psi_h(w):=\Phi'\big(\big[\Phi^{-1}(w)\big]_h\big)\big(\xi^2\big)'\big(\Phi\big([\Phi^{-1}(w)]_h\big)\big)\zeta^2\] as test function in 7 and going to the limit, we find that with \(\Psi(w)\) as described in 30 , we have \[\begin{align} \label{eq:log-est-0} \int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_0)\big)\zeta^2=\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_1)\big)\zeta^2&+\iint\limits_{Q_{r}}\nabla w\cdot\nabla \Psi(w)+\iint\limits_{Q_{r}}a\Phi^{-1}(w)\cdot\nabla\Psi(w)\nonumber\\&-\iint\limits_{Q_{r}}b\big(\cdot,\cdot,\Phi^{-1}(w)\big)\Psi(w)=:\tilde{I}_1+\tilde{I}_2+\tilde{I}_3+\tilde{I}_4. \end{align}\tag{33}\] To estimate \(\tilde{I}_2\) we draw on 31 , the nonnegativity of \(\Phi',\Phi^{-1},\zeta,\xi\) and \(\xi'\) as well as Young’s inequality to find that for all \(\eta_1>0\) \[\begin{align} \label{eq:log-est-2} \tilde{I}_2&=2\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2+2\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2\nonumber\\ &+4\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta(\nabla\zeta\cdot\nabla w)\nonumber\\ &\geq 2\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2+2\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\big(1+(1-\eta_1)\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2\nonumber\\&-\frac{2}{\eta_1}\!\iint\limits_{Q_{r}}\Phi'\big(\Phi^{-1}(w)\big)\xi(w)|\nabla\zeta|^2. \end{align}\tag{34}\] Similarly, relying on the nonnegativity of \(\Phi''\) as entailed by the assumed convexity and three applications of Young’s inequality, we estimate \(\tilde{I}_3\) to obtain that for all \(\eta_2,\eta_3,\eta_4>0\) we have \[\begin{align} \label{eq:log-est-3} \tilde{I}_3&=2\iint\limits_{Q_{r}}\!\Phi^{-1}(w)\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2(a\cdot\nabla w)\nonumber\\ &+2\iint\limits_{Q_{r}}\!\Phi^{-1}(w)\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2(a\cdot\nabla w)\nonumber\\ &+4\iint\limits_{Q_{r}}\!\Phi^{-1}(w)\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta(a\cdot\nabla\zeta)\nonumber\\ &\geq -2\eta_2\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2-\frac{1}{2\eta_2}\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)\big(\Phi^{-1}(w)\big)^2}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|a|^2\nonumber\\ &-2\eta_3\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2-8\eta_4\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\xi(w)|\nabla\zeta|^2\nonumber\\ &-\iint\limits_{Q_{r}}\!\big(\Phi^{-1}(w)\big)^2\Phi'\big(\Phi^{-1}(w)\big)\Big(\frac{1}{2\eta_3}+\Big(\frac{1}{2\eta_3}+\frac{1}{2\eta_4}\Big)\xi(w)\Big)\big(\xi'(w)\big)^2\zeta^2|a|^2. \end{align}\tag{35}\] Moreover, we clearly have \[\begin{align} \label{eq:log-est-4} \tilde{I}_4\geq -2\iint\limits_{Q_{r}}\big|b(\cdot,\cdot,\Phi^{-1}(w))\big|\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta^2, \end{align}\tag{36}\] so that by combining 33 –36 we conclude that for all \(\eta_1,\eta_2,\eta_3,\eta_4>0\) \[\begin{align} &\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_1)\big)\zeta^2+2(1-\eta_2)\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2\\ &+2\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\big(1-\eta_3+(1-\eta_1-\eta_3)\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2\\ \leq\;&\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_0)\big)\zeta^2+\Big(\frac{2}{\eta_1}+8\eta_4\Big)\!\iint\limits_{Q_{r}}\Phi'\big(\Phi^{-1}(w)\big)\xi(w)|\nabla\zeta|^2 \\& +\frac{1}{2\eta_2}\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)\big(\Phi^{-1}(w)\big)^2}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|a|^2\\ &+\iint\limits_{Q_{r}}\!\big(\Phi^{-1}(w)\big)^2\Phi'\big(\Phi^{-1}(w)\big)\Big(\frac{1}{2\eta_3}+\big(\frac{1}{2\eta_3}+\frac{1}{2\eta_4}\big)\xi(w)\Big)\big(\xi'(w)\big)^2\zeta^2|a|^2\\ &+2\iint\limits_{Q_{r}}\big|b(\cdot,\cdot,\Phi^{-1}(w))\big|\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta^2. \end{align}\] Letting \(\eta_1=\frac{1}{2}\), \(\eta_2=1\), \(\eta_3=\frac{1}{2}\) and \(\eta_4=\frac{1}{4}\) this entails that \[\begin{align} \label{eq:log-est-5} &\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_1)\big)\zeta^2\nonumber\\\leq\;&\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_0)\big)\zeta^2+6\iint\limits_{Q_{r}}\!\Phi'\big(\Phi^{-1}(w)\big)\xi(w)|\nabla\zeta|^2\nonumber +\frac{1}{2}\iint\limits_{Q_{r}}\!\frac{\Phi''\big(\Phi^{-1}(w)\big)\big(\Phi^{-1}(w)\big)^2}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|a|^2\nonumber \\& +3\iint\limits_{Q_{r}}\!\big(\Phi^{-1}(w)\big)^2\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|a|^2\nonumber \\& +2\iint\limits_{Q_{r}}\!\big|b(\cdot,\cdot,\Phi^{-1}(w))\big|\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta^2. \end{align}\tag{37}\] To further treat the last four integrals on the right-hand side and establish a connection to suitable superlevel sets, we observe that by our construction of \(\xi\) we have \[\begin{align} \label{eq:log-est-7} \xi(w)=0\text{ on }\big\{w\leq L-K\big\}\subseteq \big\{w\leq L-(1-\delta)K\big\}. \end{align}\tag{38}\] Furthermore, we note that since \(w\leq M\) the fact that both \(\Phi^{-1}\) and \(\Phi'\) are monotonically increasing entails that \[\begin{align} \label{eq:log-est-5461} \Phi'\big(\Phi^{-1}(w)\big)\leq \Phi'\big(\Phi^{-1}(M)\big)\quad\text{and}\quad\big(\Phi^{-1}(w)\big)^2\Phi'\big(\Phi^{-1}(w)\big)\leq \big(\Phi^{-1}(M)\big)^2\Phi'\big(\Phi^{-1}(M)\big) \end{align}\tag{39}\] and from Lemma 1 we find that since \(\Phi^{-1}(w)\leq \Phi^{-1}(M)\) on \(Q_r\) there is \(C_1=C_1(\Phi,M)\) such that \(\Phi''(\Phi^{-1}(w))\Phi^{-1}(w)\leq C_1\Phi'(\Phi^{-1}(w))\), so that we may estimate \[\begin{align} \label{eq:log-est-5462} \frac{\Phi''\big(\Phi^{-1}(w)\big)\big(\Phi^{-1}(w)\big)^2}{\Phi'\big(\Phi^{-1}(w)\big)}\leq C_1\Phi^{-1}(M)=:C_2(\Phi,M)\quad\text{on }Q_r. \end{align}\tag{40}\] Hence, aggregating 37 –40 and drawing on the estimates \(\xi\leq\ln(\tfrac{1}{\delta})\), \(\xi'\leq\frac{1}{\delta K}\) of 29 as well as the properties of \(\zeta\), we obtain \(C_3=C_3(\Phi,M)>0\) satisfying \[\begin{align} \int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_1)\big)\zeta^2\leq&\int_{\mathcal{K}_r}\xi^2\big(w(\cdot,T_0)\big)\zeta^2+6\ln\left(\tfrac{1}{\delta}\right)\iint\limits_{Q_r}\Phi'\big(\Phi^{-1}(w)\big)|\nabla\zeta|^2\\ &+C_3\left(\frac{\ln(\frac{1}{\delta})}{\delta K}+\frac{\left(1+\ln(\frac{1}{\delta})\right)}{\delta^2K^2}\right)\iint\limits_{Q_r\cap\{w>L-K\}}\!\!|a|^2+C_3\frac{\ln(\frac{1}{\delta})}{\delta K}\iint\limits_{Q_r\cap\{w>L-K\}}\!\!|\hat{b}|. \end{align}\] Herein, we have in view of Hölder’s inequality that \[\begin{align} &\iint\limits_{Q_r\cap\{w>L-K\}}\!\!|a|^2\leq \|a\|_{L^{2q_1\;\!}\!((0,T);L^{2q_2\;\!}\!\left(\Omega;\mathbb{R}^N\right))}\Big(\int_{T_0}^{T_1}\big|A_{L-K,r}^+(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \intertext{and} &\iint\limits_{Q_r\cap\{w>L-K\}}\!\!|\hat{b}|\leq \|\hat{b}\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2}(\Omega)\right)}\Big(\int_{T_0}^{T_1}\big|A_{L-K,r}^+(t)\big|^{\frac{q_1(q_2-1)}{q_2(q_1-1)}}\Big)^\frac{q_1-1}{q_1}, \end{align}\] so that letting \(C=C(\Phi,M,q_1,q_2,a,\hat{b}):=C_3\big(\|a\|_{L^{2q_1\;\!}\!((0,T);L^{2q_2\;\!}\!\left(\Omega;\mathbb{R}^N\right))}+\|\hat{b}\|_{L^{q_1\;\!}\!\left((0,T);L^{q_2}(\Omega)\right)}\big)>0\) yields 32 , completing the proof. ◻

Proof:. Noticing that for \(s\leq K\) we have \(\Lambda^+(s)=0\) and \((s-K)_+=0\), the inequalities in 41 are obviously valid for \(s\leq K\), so we are left with verifying them for \(s\in(K,\mu^+]\). Similarly, for \(s\geq K\) we have \(\Lambda^-(s)=0\) and \((s-K)_-=0\), so clearly 42 and 43 hold for \(s\geq K\) and we only have to consider 42 and 43 for \(s\in[\mu^-,K)\).

By change of variables \(\sigma=\Phi(\tilde{\sigma})\), we find that \[\begin{align} \Lambda^+(s)=\int_{K}^{s}\!\big(\sigma-K\big)_+\big(\Phi^{-1}\big)'(\sigma)\mathop{\mathrm{d\!}}\sigma\quad\text{for }s\in(K,\mu^+] \end{align}\] and \[\begin{align} \Lambda^-(s)=\int_{s}^{K}\!\big(\sigma-K\big)_-\big(\Phi^{-1}\big)'(\sigma)\mathop{\mathrm{d\!}}\sigma\quad\text{for }s\in[\mu^-,K). \end{align}\] Due to \(\Phi\) being convex, \(\Phi'\) is monotonically increasing and hence we conclude from the monotonically increasing property of \(\Phi^{-1}\) and the inversion rule that \[\frac{1}{\Phi'\big(\Phi^{-1}(\mu^+)\big)}\leq \big(\Phi^{-1}\big)'(\sigma)\leq\frac{1}{\Phi'\big(\Phi^{-1}(K)\big)}\quad\text{ for all }\sigma\in(K,\mu^+].\] Accordingly, we find that \[\begin{align} \Lambda^+(s)\geq\frac{1}{\Phi'\big(\Phi^{-1}(\mu^+)\big)}\int_{K}^{s}(\sigma-K)_+\mathop{\mathrm{d\!}}\sigma=\frac{(s-K)_+^2}{2\Phi'\big(\Phi^{-1}(\mu^+)\big)}\quad\text{for all }s\in(K,\mu^+] \end{align}\] as well as \[\begin{align} \Lambda^+(s)\leq \frac{1}{\Phi'\big(\Phi^{-1}(K)\big)}\int_K^s(\sigma-K)_+\mathop{\mathrm{d\!}}\sigma=\frac{(s-K)_+^2}{2\Phi^{-1}\big(\Phi(K)\big)}\quad\text{for all }s\in(K,\mu^+]. \end{align}\] In a similar fashion, we conclude from \(\big(\Phi^{-1}\big)'(\sigma)\geq\frac{1}{\Phi'(\Phi^{-1}(K))}\) for all \(\sigma\in[\mu^-,K)\) that \[\begin{align} \Lambda^-(s)\geq \frac{(s-K)_-^2}{2\Phi'\big(\Phi^{-1}(K)\big)}\quad\text{for all }s\in[\mu^-,K). \end{align}\] Let us turn to the upper bound on \(\Lambda^-\) in 42 . Here, we note that \((\sigma-K)_-\leq (s-K)_-\) for all \(\sigma\in[s,K)\) and thus \[\begin{align} \label{eq:proof-lambda-1} \Lambda^-(s)&\leq (s-K)_-\int_s^{K}\big(\Phi^{-1}\big)'(\sigma)\mathop{\mathrm{d\!}}\sigma=(s-K)_-\Big(\Phi^{-1}(K)-\Phi^{-1}(s)\Big). \end{align}\tag{44}\] In view of \(\Phi(0)=0\), we can make use of the convexity of \(\Phi\) and the fact that \(\Phi^{-1}\) is monotonically increasing to estimate \[\begin{align} \frac{K}{\Phi^{-1}(K)}=\frac{\Phi\big(\Phi^{-1}(K)\big)-\Phi\big(\Phi^{-1}(0)\big)}{\Phi^{-1}(K)-\Phi^{-1}(0)}\leq \frac{\Phi\big(\Phi^{-1}(K)\big)-\Phi\big(\Phi^{-1}(s)\big)}{\Phi^{-1}(K)-\Phi^{-1}(s)}=\frac{K-s}{\Phi^{-1}(K)-\Phi^{-1}(s)} \end{align}\] for \(K> s\geq \mu^-\geq 0\), so that actually \[\begin{align} \Phi^{-1}(K)-\Phi^{-1}(s)\leq \frac{\Phi^{-1}(K)}{K}(K-s)\quad\text{for }\mu^-\leq s< K. \end{align}\] Plugging this into 44 , we obtain for \(K> s\geq \mu^-\) that \[\Lambda^-(s)\leq \frac{\Phi^{-1}(K)}{K}(s-K)_-(K-s)= \frac{\Phi^{-1}(K)}{K}(s-K)_-^2.\] The estimate 43 in the special case when \(\mu^->0\) follows from arguments along the line of the estimate from above for \(\Lambda^+\), utilizing that in this case we have \(0<\Phi'\big(\Phi^{-1}(\mu^-)\big)\leq \Phi'(\Phi^{-1}(s))\) for all \(s\geq \mu^->0\) again by the monotonicity properties of \(\Phi\) and \(\Phi^{-1}\). ◻

Proof:. Setting \(\hat{\mu}_-:=\mathop{\mathrm{ess\,inf}}_{Q_{2r}^{\theta}}w\) and \(\mathop{\mathrm{ess\, sup}}_{Q_{2r}^{\theta}}w=:\hat{\mu}_+\), we first note that the assumption \(\vartheta\geq\theta\) entails that \[\begin{align} Q_{2r}^\theta\subseteq Q_{2r}^{\vartheta}\quad\text{and hence}\quad \mu^-\leq\hat{\mu}_-\leq\hat{\mu}_+\leq\mu^+. \end{align}\] Our aim is to employ the geometric convergence Lemma 4 for a suitable sequence of sublevel sets of nested cylinders. To obtain the recursive inequality required by said lemma, we will use the local energy estimate presented in Lemma 5 for \(\Lambda^-\). Let us start by preparing the sequence of levels as well as the nested cylinders. Set \(r_0=r\) and \(k_0=\mu^-+k>0\) and for \(i\in\mathbb{N}\) define \[r_i=\frac{r}{2}+\frac{r}{2^{i+1}}\quad\text{and}\quad k_i=\mu^-+\frac{k}{2}+\frac{k}{2^{i+1}}.\] Evidently, for all \(i\in\mathbb{N}_0\) we have \(r_i\in(\frac{r}{2},r]\) and \(k_i\in(\mu^-+\frac{k}{2},\mu^-+k]\) and \[\begin{align} \label{eq:first-alt-1-rinfty} r_\infty:=\lim_{i\to\infty}r_i=\frac{r}{2}\quad\text{and}\quad k_\infty:=\lim_{i\to\infty} k_i=\mu^-+\frac{k}{2}. \end{align}\tag{48}\] We introduce \(Q_i:=Q^{\theta}_{r_i}=\mathcal{K}_{r_i}\times(t_0-\theta r_i^2,t_0]\) and fix a sequence of smooth cutoff functions \(\psi_i\in C^{\infty}_0\big(\Omega_T\big)\) such that \[\begin{align} \label{eq:first-alt-1-psi-est} &\psi_i\equiv 1\;\text{ in }\;Q_{i+1},\quad \psi_i\equiv 0\;\text{ in }\;\Omega_T\setminus \overline{Q_i},\;\text{ and }\;\psi_{\vert\partial_p Q_i}=0\nonumber\\&\text{with }\;|\nabla\psi_i|\leq \frac{2^{i+3}}{r},\;\text{ as well as }\;\big|(\psi_{i})_t\big|\leq\frac{2^{2(i+2)}}{\theta r^2}\;\text{ on }Q_i. \end{align}\tag{49}\] Since our estimations for \[J_i:=\sup_{t_0-\theta r_i^2\leq t\leq t_0}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,t)-k_i\big)_-^2\psi^2_i(\cdot,t)+\frac{1}{\theta_{k_i}}\iint\limits_{Q_{i}}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\] differ slightly in each of the cases for \(\theta\), let us briefly split our argument up in order to verify that the specific choice for \(\theta\) does not impact the constants appearing below.

First we consider the instance where \(\theta=\theta_k\). We recall that according to Lemma 7 for each \(i\in\mathbb{N}_0\) we have \[\Lambda^-_i(w):=\Lambda^-(w,k_i)\leq\frac{\Phi^{-1}(k_i)}{k_i}(w-k_i)_-^2\quad\text{ and }\quad\Lambda_i^-(w)\geq \frac{1}{2\Phi'(\Phi^{-1}(k_i))}(w-k_i)_-^2\] a.e. in \(Q_i\). Hence, we find from employing Lemma 5 for \(\Lambda_i^-(w)\) inside the cylinders \(Q_i\) that there is \(\Gamma_1=\Gamma_1(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that for all \(i\in\mathbb{N}_0\) we have \[\begin{align} &\sup_{t_0-\theta r_i^2\leq t\leq t_0}\frac{1}{\Phi'(\Phi^{-1}(k_i))}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,t)-k_i\big)_-^2\psi^2_i(\cdot,t)+\iint\limits_{Q_{i}}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\\ \leq\;&\Gamma_1\frac{\Phi^{-1}(k_i)}{k_i}\iint\limits_{Q_{i}}(w-k_i)_-^2\psi_i|(\psi_i)_t|+\Gamma_1\iint\limits_{Q_{i}}(w-k_i)_-^2|\nabla\psi_i|^2+\Gamma_1\Big(\int_{t_0-\theta r_i^2}^{t_0}\big|A_{k_i,r_i}^-(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}. \end{align}\] Multiplying this inequality by \(\theta_{k_i}^{-1}=\Phi'\big(\Phi^{-1}(k_i)\big)\) and recalling that \(k_i\leq k_0= \mu^-+k\leq 2M\) for all \(i\in\mathbb{N}_0\) and Lemma 1 ensure the existence of \(C_1=C_1(\Phi,M)>0\) such that \[\begin{align} \Phi'\big(\Phi^{-1}(k_i)\big)\Phi^{-1}(k_i)\leq C_1 \Phi\big(\Phi^{-1}(k_i)\big)=C_1 k_i. \end{align}\] we find \(\Gamma_2=\Gamma_2(\Phi,M,q_1,q_2,a,\hat{b})>0\) satisfying \[\begin{align} J_i\leq\Gamma_2\iint\limits_{Q_{i}}(w-k_i)_-^2\psi_i|(\psi_i)_t|+\frac{\Gamma_2}{\theta_{k_0}}\iint\limits_{Q_{i}}(w-k_i)_-^2|\nabla\psi_i|^2+\Gamma_2\Big(\int_{t_0-\theta r_i^2}^{t_0}\!\!\big|A_{k_i,r_i}^-(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] for all \(i\in\mathbb{N}_0\), where we additionally used \(\theta_{2M}\leq\theta_{k_0}\leq\theta_{k_i}\) for all \(i\in\mathbb{N}_0\) entailed by the monotonicity properties of \(\Phi'\) and \(\Phi^{-1}\).

For the case \(\theta=\theta_{\mu^-}\) we note – again by monotonicity of \(\Phi'\) and \(\Phi^{-1}\) – that necessarily \(\mu^-\geq k>0\), so that the special case of Lemma 7 becomes applicable, yielding \[\begin{align} \Lambda^-_i(w):=\Lambda^-(w,k_i)\leq\frac{1}{2\Phi'(\Phi^{-1}(\mu^-))}(w-k_i)_-^2\quad\text{ and }\quad\Lambda_i^-(w)\geq \frac{1}{2\Phi'(\Phi^{-1}(k_i))}(w-k_i)_-^2 \end{align}\] a.e. in \(Q_i\). Hence, this time we find from employing Lemma 5, multiplying by \(\theta_{k_i}^{-1}=\Phi'\big(\Phi^{-1}(k_i)\big)\) and again using \(k_i\leq 2M\), that there is \(\Gamma_3=\Gamma_3(\Phi,M,q_1,q_2,a,\hat{b})>0\) satisfying \[\begin{align} J_i\leq\Gamma_3\frac{\theta_{\mu^-}}{\theta_{k_i}}\iint\limits_{Q_{i}}(w-k_i)_-^2\psi_i|(\psi_i)_t| &+\frac{\Gamma_3}{\theta_{k_0}}\iint\limits_{Q_{i}}(w-k_i)_-^2|\nabla\psi_i|^2+\Gamma_3\Big(\int_{t_0-\theta r_i^2}^{t_0}\!\!\big|A_{k_i,r_i}^-(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] for all \(i\in\mathbb{N}_0\). Since \(\mu^-\geq k\) we have \(k_i\leq k_0=\mu^-+k\leq 2\mu^-\), so that \(\theta_{k_i}\geq \theta_{2\mu^-}\). Accordingly, drawing on the fact that \(\Phi^{-1}\) is subadditive due to its assumed concavity and \(\Phi^{-1}(0)=0\), we obtain from Lemma 1 that there is \(C_2=C_2(\Phi,M)>0\) satisfying \[\begin{align} \frac{\theta_{\mu^-}}{\theta_{k_i}}&\leq \frac{\theta_{\mu^-}}{\theta_{2\mu^-}}=\frac{\Phi'(\Phi^{-1}(2\mu^-))}{\Phi'(\Phi^{-1}(\mu^-))}\leq\frac{\Phi'(2\Phi^{-1}(\mu^-))}{\Phi'(\Phi^{-1}(\mu^-))}\leq C_2 \quad\text{for all }i\in\mathbb{N}_0. \end{align}\] Hence, letting \(\Gamma_4=\Gamma_4(\Phi,M,q_1,q_2,a,\hat{b}):=\max\{\Gamma_2,\Gamma_3C_2,\Gamma_3\}>0\) we find that \[\begin{align} J_i\leq\Gamma_4\iint\limits_{Q_{i}}(w-k_i)_-^2\psi_i|(\psi_i)_t| &+\frac{\Gamma_4}{\theta_{k_0}}\iint\limits_{Q_{i}}(w-k_i)_-^2|\nabla\psi_i|^2+\Gamma_4\Big(\int_{t_0-\theta r_i^2}^{t_0}\!\!\big|A_{k_i,r_i}^-(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] holds for all \(i\in\mathbb{N}_0\) irrespective of a specific case for \(\theta\). In light of the fact that by the definition of \(k_i\) we have \((w-k_i)_-\leq (w-k_0)_-\leq(\hat{\mu}_--\mu^--k)_-\leq k\) a.e. in \(Q_{i}\), we derive from the estimates provided by 49 that by writing \(A_i^-:=A_{k_i,r_i}^-\), \(\widehat{q_1}:=\frac{2q_1(1+\kappa)}{q_1-1}\) and \(\widehat{q_2}:=\frac{2q_2(1+\kappa)}{q_2-1}\) we obtain \[\begin{align} \label{eq:first-alt-1-eq0} J_i\leq \Gamma_4\frac{2^{2i+4}}{\theta r^2}k^2\big|A_{i}^-\big|+\Gamma_4\frac{2^{2i+6}}{\theta_{k_0}r^2}k^2\big|A_{i}^-\big|+\Gamma_4\Big(\int_{t_0-\theta r_i^2}^{t_0}\big|A_{k_i,r_i}^-(t)\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\Big)^{\frac{2(1+\kappa)}{\widehat{q_1}}} \end{align}\tag{50}\] is valid for all \(i\in\mathbb{N}_0\). Now, we introduce a change in variable by setting \(\bar{t}:=\frac{t-t_0}{\theta}\) and correspondingly denote \(\bar{Q}_i:=\mathcal{K}_{r_i}\times(-r_i^2,0]\), \(\bar{A}_i^-:=\bar{Q}_i\cap\big\{(x,\bar{t})\in\Omega\times(-\frac{t_0}{\theta},\frac{T-t_0}{\theta}):\big(w(x,\bar{t})-k_i\big)_->0\big\}\) and \(\bar{A}^-_{i}(\bar{t}):=A_{k_i,r_i}^-(\theta\bar{t}+t_0)\). This way we have \(|A_i^-|=\theta|\bar{A}_i^-|\) and we conclude from 50 that \[\begin{align} \label{eq:first-alt-1-eq1} &\sup_{-r_i^2\leq\,\bar{t}\,\leq 0}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,\bar{t})-k_i\big)_-^2\psi_i^2(\cdot,\bar{t})+\frac{\theta}{\theta_{k_i}}\iint\limits_{\bar{Q}_i}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\nonumber\\ &\leq\Gamma_4\frac{2^{2i+4}}{r^2}k^2 \big|\bar{A}_i^-\big|+\Gamma_4\frac{\theta}{\theta_{k_0}}\frac{2^{2i+6}}{r^2}k^2\big|\bar{A}_i^-\big|+\Gamma_4\theta^{{\frac{2(1+\kappa)}{\widehat{q_1}}}}\Big(\int_{-r_i^2}^0\big|\bar{A}_{i}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}{\bar{t}}\Big)^{\frac{2(1+\kappa)}{\widehat{q_1}}} \end{align}\tag{51}\] for all \(i\in\mathbb{N}_0\). If \(\mu^-\leq k\), we have \(\frac{k}{2}\leq k_i\leq k_0\leq 2k\) and the subadditivity of \(\Phi^{-1}\) then implies that \[\begin{align} \Phi'\big(\Phi^{-1}(k_0)\big)\leq\Phi'\big(\Phi^{-1}(2k)\big)\leq\Phi'\big(2\Phi^{-1}(k)\big)\quad\text{and}\quad \Phi'\big(\Phi^{-1}(k)\big)\leq\Phi'\big(2\Phi^{-1}(\tfrac{k}{2})\big). \end{align}\] Accordingly, in this case we find from \(\theta=\theta_k\), \(k\leq \mu^+\leq M\) and 16 of Lemma 1 that there is some \(C_3=C_3(M)>1\) for which \[\begin{align} \frac{\theta}{\theta_{k_0}}=\frac{\Phi'\big(\Phi^{-1}(k_0)\big)}{\Phi'\big(\Phi^{-1}(k)\big)}\leq\frac{\Phi'\big(2\Phi^{-1}(k)\big)}{\Phi'\big(\Phi^{-1}(k)\big)}\leq C_3\quad\text{and}\quad \frac{1}{C_3}\leq\frac{\Phi'\big(\Phi^{-1}(\tfrac{k}{2})\big)}{\Phi'\big(2\Phi^{-1}(\tfrac{k}{2})\big)}\leq \frac{\Phi'\big(\Phi^{-1}(k_i)\big)}{\Phi'\big(\Phi^{-1}(k)\big)}=\frac{\theta}{\theta_{k_i}} \end{align}\] for all \(i\in\mathbb{N}_0\). Similarly, when \(\mu^->k\) we have \(\mu^-< k_i \leq k_0< 2\mu^-\) and thus \[\begin{align} \Phi'\big(\Phi^{-1}(k_0)\big)\leq\Phi'\big(2\Phi^{-1}(\mu^-)\big)\quad\text{and}\quad\Phi'\big(\Phi^{-1}(\mu^-)\big)\leq\Phi'\big(\Phi^{-1}(k_i)\big), \end{align}\] which together with Lemma 1 yields \(C_4=C_4(M)> 1\) such that \[\begin{align} \frac{\theta}{\theta_{k_0}}\leq \frac{\Phi'\big(2\Phi^{-1}(\mu^-)\big)}{\Phi'\big(\Phi^{-1}(\mu^-)\big)}\leq C_4\quad\text{and}\quad \frac{1}{C_4}\leq\frac{\Phi'\big(\Phi^{-1}(\mu^-)\big)}{\Phi'\big(\Phi^{-1}(\mu^-)\big)}\leq\frac{\theta}{\theta_{k_i}}\quad\text{for all }i\in\mathbb{N}_0. \end{align}\] Hence, we obtain from 51 that in both cases there is \(\Gamma_5=\Gamma_5(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \label{eq:first-alt-1-eq1465} \bar{J}_i&:=\sup_{-r_i^2\leq\,\bar{t}\,\leq 0}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,\bar{t})-k_i\big)_-^2\psi_i^2(\cdot,\bar{t})+\iint\limits_{\bar{Q}_i}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\nonumber\\ &\leq \Gamma_5\frac{2^{2i}k^2}{r^2}\big|\bar{A}_i^-\big|+\Gamma_5\theta^{\frac{q_1-1}{q_1}}\Big(\int_{-r_i^2}^0\big|\bar{A}_{i}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}{\bar{t}}\Big)^{\frac{2(1+\kappa)}{\widehat{q_1}}}\quad\text{for all }i\in\mathbb{N}_0. \end{align}\tag{52}\] In order to relate this to the smaller cylinder in step \(i+1\), we estimate \(\bar{J}_i\) from below using Corollary 3. Since \(\psi_i=1\) on \(\bar{Q}_{i}\) and \(\psi_i\geq0\) on \(\bar{Q}_{i}\setminus \bar{Q}_{i+1}\), we have \[\begin{align} \iint\limits_{\bar{Q}_{i+1}}(w-k_i)_-^2\leq \iint\limits_{\bar{Q}_i}(w-k_i)_-^2\psi_i^2 \end{align}\] for all \(i\in\mathbb{N}_0\). From Corollary 3 we thereby find \(C_5>0\) and \(C_6=C_6(M)>0\) such that \[\begin{align} \label{eq:first-alt-1-eq2} \iint\limits_{\bar{Q}_{i+1}}(w-k_i)_-^2\nonumber\leq\;& C_5\big|\bar{A}_{i}^-\big|^\frac{2}{N+2}\Big(\mathop{\mathrm{ess\, sup}}_{\bar{t}\in(-r_i^2,0)}\|(w-k_i)_-\psi_i\|_{L^{2\;\!}\!(\mathcal{K}_{r_i})}+\Big(\iint\limits_{\bar{Q}_{i}}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\Big)^\frac{1}{2}\Big)^2\nonumber\\ \leq\;& C_6\big|\bar{A}_{i}^-\big|^\frac{2}{N+2}\Big(\mathop{\mathrm{ess\, sup}}_{-r_i^2\leq \bar{t}\leq 0}\int_{\mathcal{K}_{r_i}}(w-k_i)_-^2\psi_i^2+\iint\limits_{\bar{Q}_i}\big|\nabla\big((w-k_i)_-\psi_i\big)\big|^2\Big)\nonumber\\ \leq\;&C_6\big|\bar{A}_{i}^-\big|^\frac{2}{N+2} \bar{J}_i\quad\text{for all }i\in\mathbb{N}_0. \end{align}\tag{53}\] Moreover, note that \((w-k_i)_-=-w+k_i\geq k_i-k_{i+1}\) whenever \(w\leq k_{i+1}<k_i\), and thus \((w-k_i)_-^2\geq (k_i-k_{i+1})^2\) a.e. on \(\bar{A}_{i+1}^-\). Accordingly, since \(k_i-k_{i+1}=\frac{k}{2^{i+2}}\), \[\begin{align} \iint\limits_{\bar{Q}_{i+1}}(w-k_i)_-^2\geq (k_i-k_{i+1})^2\big|\bar{A}_{i+1}^-\big|=\frac{k^2}{2^{2i+4}} \big|\bar{A}_{i+1}^-\big|, \end{align}\] which upon combination with 52 and 53 implies that there is \(\Gamma_6=\Gamma_6(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \big|\bar{A}_{i+1}^-\big|&\leq\frac{C_6 2^{2i+4}}{k^2}\big|\bar{A}_{i}^-\big|^\frac{2}{N+2} \bar{J}_i\\ &\leq \Gamma_6\frac{2^{4i}}{r^2}\big|\bar{A}_{i}^-\big|^{1+\frac{2}{N+2}}+\Gamma_6\frac{2^{2i}}{k^2}\theta^{\frac{q_1-1}{q_1}}\big|\bar{A}_{i}^-\big|^\frac{2}{N+2}\Big(\int_{-r_i^2}^{0}\big|\bar{A}_{i}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}\bar{t}\Big)^{\frac{2(1+\kappa)}{\widehat{q_1}}} \end{align}\] holds for all \(i\in\mathbb{N}_0\). Dividing this by \(r^{N+2}\) and introducing the quantities \[\bar{Y}_i:=\frac{\big|\bar{A}_{i}^-\big|}{r^{N+2}}\;\text{ as well as }\;\bar{Z}_i:=\frac{1}{r^N}\Big(\int_{-r_i^2}^{0}\big|\bar{A}_{i}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}\bar{t}\Big)^{\frac{2}{\widehat{q_1}}},\quad i\in\mathbb{N}_0,\] we therefore have \[\begin{align} \label{eq:first-alt-1-Y} \bar{Y}_{i+1}&\leq \Gamma_6 2^{4i}\Big(\bar{Y}_i^{1+\frac{2}{N+2}}+\frac{r^{N\kappa}}{k^{2}}\theta^{\frac{q_1-1}{q_1}}\bar{Y}_i^\frac{2}{N+2}\bar{Z}_i^{1+\kappa} \Big)\quad\text{for all }i\in\mathbb{N}_0. \end{align}\tag{54}\] Regarding \(\bar{Z}_{i+1}\), we note that \((w-k_i)_-^2\geq (k_i-k_{i+1})^2\) whenever \(w<k_{i+1}\), and therefore we may estimate for \(i\in\mathbb{N}_0\) \[\begin{align} &r^N(k_i-k_{i+1})^2 \bar{Z}_{i+1}\\ =\;&(k_i-k_{i+1})^2\Big(\int_{-r_{i+1}^2}^{0}\big|\bar{A}_{i+1}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}\bar{t}\Big)^{\frac{2}{\widehat{q_1}}}\\ \leq\;&\Big(\int_{-r_{i+1}^2}^{0}\Big(\int_{\mathcal{K}_{r_{i+1}}}\big(w(x,\bar{t})-k_i\big)_-^{\widehat{q_2}}\Big)^{\frac{\widehat{q_1}}{\widehat{q_2}}}\Big)^{\frac{2}{\widehat{q_1}}}\\ \leq\;&\Big(\int_{-r_{i}^2}^{0}\Big(\int_{\mathcal{K}_{r_{i}}}\big(w(x,\bar{t})-k_i\big)_-^{\widehat{q_2}}\psi_i^{\widehat{q_2}}(\cdot,\bar{t})\Big)^{\frac{\widehat{q_1}}{\widehat{q_2}}}\Big)^{\frac{2}{\widehat{q_1}}}\\ =\;&\big\|\big((w-k_i)_-\psi_i\big)(x,\bar{t})\big\|_{L^{\widehat{q_1}\;\!}\!((-r_i^2,0);L^{\widehat{q_2}\;\!}\!(\mathcal{K}_{r_i}))}^2. \end{align}\] Here, our assumptions on \(q_1,q_2\) and \(\kappa\) ensure that the embedding Lemma 3 with \(p=2\) applies, yielding \(C_7>0\) satisfying \[\begin{align} r^N(k_i-k_{i+1})^2 \bar{Z}_{i+1}\leq C_7 \bar{J}_i\quad\text{for }i\in\mathbb{N}_0, \end{align}\] which in view of \(k^2=2^{2i+4}(k_i-k_{i+1})^2\) and 52 entails that with some \(\Gamma_7=\Gamma_7(\Phi,M,q_1,q_2,a,\hat{b})>0\), \[\begin{align} \label{eq:first-alt-1-Z} \bar{Z}_{i+1}&\leq \frac{2^{2i+4}C_7}{r^Nk^2}\Big(\Gamma_5 \frac{2^{2i}k^2}{r^2}r^{N+2}\bar{Y}_i+\Gamma_5r^{N+N\kappa}\theta^{\frac{q_1-1}{q_1}} \bar{Z}_i^{1+\kappa}\Big)\leq 2^{4i}\Gamma_7 \Big(\bar{Y}_{i}+\frac{r^{N\kappa}}{k^{2}}\theta^{\frac{q_1-1}{q_1}}\bar{Z}_{i}^{1+\kappa}\Big) \end{align}\tag{55}\] for \(i\in\mathbb{N}_0\). We conclude from 54 and 55 that with \(\Gamma_8=\Gamma_8(\Phi,M,q_1,q_2,a,\hat{b}):=\max\big\{\Gamma_6,\Gamma_7 \big\}+1>1\) we have \[\begin{align} \bar{Y}_{i+1}\leq \Gamma_8 2^{4i}\Big(\bar{Y}_i^{1+\frac{2}{N+2}}+\frac{r^{N\kappa}}{k^{2}}\theta^{\frac{q_1-1}{q_1}}\bar{Y}_i^\frac{2}{N+2} \bar{Z}_i^{1+\kappa}\Big)\quad\text{and}\quad \bar{Z}_{i+1}\leq \Gamma_8 2^{4i}\Big(\bar{Y}_i+\frac{r^{N\kappa}}{k^{2}}\theta^{\frac{q_1-1}{q_1}}\bar{Z}_i^{1+\kappa}\Big) \end{align}\] for \(i\in\mathbb{N}_0\). Assuming \(k^{2}\theta^{-\frac{q_1-1}{q_1}}>r^{N\kappa}\), we obtain recursive inequalities precisely of the form requested by Lemma 4 and are left with checking the condition on \(\bar{Y}_0+\bar{Z}_0^{1+\kappa}\). Recall \(r_0=r\) and \(k_0=\mu^-+k\). Then, first assuming \(\widehat{q_1}\geq\widehat{q_2}\), we use \[\frac{2(1+\kappa)}{\widehat{q_1}}\left(\frac{\widehat{q_1}}{\widehat{q_2}}-1\right)=2(1+\kappa)\left(\frac{1}{\widehat{q_2}}-\frac{1}{\widehat{q_1}}\right)=\frac{1}{q_1}-\frac{1}{q_2}\] and that \(|\bar{A}_0^-(\bar{t})|\leq |K_r|\) for all \(\bar{t}\in(-\tfrac{t_0}{\theta},\frac{T-t_0}{\theta})\) to estimate \[\begin{align} \bar{Y}_0+\bar{Z}_0^{1+\kappa}&=\frac{\big|\bar{A}_0^-\big|}{r^{N+2}}+\frac{1}{r^{N(1+\kappa)}}\Big(\int_{-r^2}^{0}\big|\bar{A}_{0}^-(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}{\bar{t}}\Big)^{\frac{2(1+\kappa)}{\widehat{q_1}}}\\ &\leq \frac{\big|\bar{A}_{\mu_-+k,r}^-\big|}{r^{N+2}}+\frac{|\mathcal{K}_r|^{\frac{1}{q_1}-\frac{1}{q_2}}}{r^{N(1+\kappa)}}\big|\bar{A}^-_{\mu^-+k,r}\big|^{\frac{2(1+\kappa)}{\widehat{q_1}}}\\ &\leq \frac{\big|\bar{A}_{\mu_-+k,r}^-\big|}{r^{N+2}}+\omega_N^{\frac{1}{q_1}-\frac{1}{q_2}}r^{\frac{N}{q_1}-\frac{N}{q_2}-N-N\kappa}\big|\bar{A}^-_{\mu^-+k,r}\big|^\frac{q_1-1}{q_1}, \end{align}\] where \(\omega_N:=2^N\) denotes the volume of the \(N\)-cube of edge length \(2\). Here, we find by the assumption 45 that \[\theta|\bar{A}^-_{\mu^-+k,r}|=|A_{\mu^-+k,r}^-|\leq\varpi_0|Q_r^{\theta}|= \omega_N\varpi_0\theta r^{N+2},\] and, since in light of 4 \(2-\frac{2}{q_1}-\frac{N}{q_2}=N\kappa\), we have \[\begin{align} \label{eq:first-alt-eq3} \bar{Y}_0+\bar{Z}_0^{1+\kappa}\leq \omega_N\varpi_0 +r^{2-\frac{2}{q_1}-\frac{N}{q_2}-N\kappa}\omega_N^{\frac{q_2-1}{q_2}} \varpi_0^{\frac{q_1-1}{q_1}}=\omega_N\varpi_0+\omega_N^{\frac{q_2-1}{q_2}}\varpi_0^{\frac{q_1-1}{q_1}}. \end{align}\tag{56}\] On the other hand, if \(\widehat{q_1}<\widehat{q_2}\) we draw on Jensen’s inequality applied to the concave function \(s\mapsto s^{{\frac{\widehat{q_1}}{\widehat{q_2}}}},\) \(s\geq 0\) to estimate \[\begin{align} \int_{-r^2}^0\big|\bar{A}_0^-(\bar{t})\big|^{{\frac{\widehat{q_1}}{\widehat{q_2}}}}\mathop{\mathrm{d\!}}\bar{t}\leq r^{2(1-{\frac{\widehat{q_1}}{\widehat{q_2}}})}\Big(\int_{-r^2}^0\big|\bar{A}_0^-(\bar{t})\big|\mathop{\mathrm{d\!}}\bar{t}\Big)^{\frac{\widehat{q_1}}{\widehat{q_2}}} =r^{2(1-{\frac{\widehat{q_1}}{\widehat{q_2}}})}\big|\bar{A}_0^-\big|^{{\frac{\widehat{q_1}}{\widehat{q_2}}}} \end{align}\] and then use \[\frac{4(1+\kappa)}{\widehat{q_1}}\left(1-\frac{\widehat{q_1}}{\widehat{q_2}}\right)=4(1+\kappa)\left(\frac{1}{\widehat{q_1}}-\frac{1}{\widehat{q_2}}\right)=\frac{2}{q_2}-\frac{2}{q_1}\] to obtain \[\begin{align} \label{eq:first-alt-eq3465} \bar{Y}_0+\bar{Z}_0^{1+\kappa}&\leq\frac{\big|\bar{A}_{\mu_-+k,r}^-\big|}{r^{N+2}}+r^{\frac{2}{q_2}-\frac{2}{q_1}-N-N\kappa}\big|\bar{A}^-_{\mu^-+k,r}\big|^\frac{q_2-1}{q_2}\nonumber\\ &\leq\omega_N\varpi_0+r^{\frac{(N+2)(q_2-1)}{q_2}+\frac{2}{q_2}-\frac{2}{q_1}-N-N\kappa}\omega_N^{\frac{q_2-1}{q_2}}\varpi_0^{\frac{q_2-1}{q_2}}=\omega_N\varpi_0+\omega_N^\frac{q_2-1}{q_2}\varpi_0^{\frac{q_2-1}{q_2}} \end{align}\tag{57}\] similarly as before. Hence, we conclude from 56 and 57 that in both of the cases \(\widehat{q_1}\geq\widehat{q_2}\) and \(\widehat{q_1}<\widehat{q_2}\), taking \(\varpi_0=\varpi_0(\Phi,M,q_1,q_2,a,\hat{b})\) so small that we have \(\varpi_0<\frac{1}{2}\) and that with \(q:=\min\big\{\frac{q_1-1}{q_1},\frac{q_2-1}{q_2}\big\}\in(0,1)\) and \(\sigma=\min\{\kappa,\frac{2}{N+2}\}=\frac{2}{N+2}\in(0,1)\) we also have \[\begin{align} \omega_N\varpi_0+\omega_N^{\frac{q_2-1}{q_2}} \varpi_0^{q}\leq \big(2\Gamma_8\big)^{-\frac{1+\kappa}{\sigma}}2^{-\frac{4(1+\kappa)}{\sigma^2}}, \end{align}\] entails \[\begin{align} \bar{Y}_0+\bar{Z}_0^{1+\kappa}\leq \big(2\Gamma_8\big)^{-\frac{1+\kappa}{\sigma}}2^{-\frac{4(1+\kappa)}{\sigma^2}}. \end{align}\] Therefore, Lemma 4 becomes applicable to say \(\bar{Y}_i\to0\) as \(i\to\infty\) and reverting to the original variable \(t\) we arrive at \[0=\frac{\big|\bar{A}_{k_\infty,r_\infty}^-\big|}{r^{N+2}}=\frac{|A_{k_\infty,r_\infty}^-|}{\theta r^{N+2}}=\frac{\omega_N|A_{k_\infty,r_\infty}^-|}{|Q_{r}^{\theta}|}.\] Recalling the values for \(k_\infty\) and \(r_\infty\) specified in 48 , we obtain 47 . ◻

Proof:. First note that since \(l\leq k\), we have \(A^+_{\mu^+-l,r}(t)\subseteq A^+_{\mu^+-k,r}(t)\) for each \(t\in[t_0-\theta r^2,t_0]\), so that it clearly suffices to verify the claim for \(l=k\).
Now, assuming to the contrary that with \(\gamma\in(0,\varpi_0)\subset(0,\frac{1}{2})\) we had \[\big|A_{\mu^+-k,r}^+(\tau)\big|>\frac{1-\varpi_0}{1-\gamma}|\mathcal{K}_r|\quad\text{for all }\tau\in\big[t_0-\theta r^2,t_0-\gamma\theta r^2\big],\] then \[\begin{align} \big|\big\{(x,t)\in Q^{\theta}_r:(w-\mu^++k)_+>0\big\}\big|&=\int_{t_0-\theta r^2}^{t_0}\big|A_{\mu^+-k,r}^+(\tau)\big|\mathop{\mathrm{d\!}}\tau\\ &\geq\int_{t_0-\theta r^2}^{t_0-\gamma\theta r^2}\big|A^+_{\mu^+-k,r}(\tau)\big|\mathop{\mathrm{d\!}}\tau\\ &>\frac{1-\varpi_0 }{1-\gamma}(1-\gamma)\theta r^2|\mathcal{K}_r|\\ &=\big(1-\varpi_0 \big)|Q^{\theta}_r|, \end{align}\] which clearly contradicts the assumption \(\big|\big\{(x,t)\in Q^{\theta}_r:(w-\mu^++k)_+>0\big\}\big|\leq \big(1-\varpi_0 \big)|Q^{\theta}_r|\). ◻

Proof:. With \(\varpi_0\) taken from Lemma 8, we set \(\gamma:=\frac{\varpi_0}{2}\) and denote by \(C_1=C_1(\Phi,M,q_1,q_2,a,\hat{b})>0\) the constant provided by Lemma 6. In view of the monotonicity of \(\Phi^{-1}\), we find from \(\mu^-<\mu^+\leq M\) and \(k\leq \mu^+\leq M\) that \(\Phi^{-1}(\mu^-)\leq \Phi^{-1}(M)\) and \(\Phi^{-1}(k)\leq \Phi^{-1}(M)\). Thus, according to Lemma 1, we can pick \(C_2=C_2(\Phi,M)>0\) such that \[\begin{align} \label{eq:exp-in-time-eq0} \frac{\Phi'\big(5\Phi^{-1}(k)\big)}{\Phi'\big(\Phi^{-1}(k)\big)}\leq C_2\quad\text{and}\quad\frac{\Phi'\big(5\Phi^{-1}(\mu^-)\big)}{\Phi'\big(\Phi^{-1}(\mu^-)\big)}\leq C_2. \end{align}\tag{60}\] Next, we define \(\lambda:=\frac{\gamma^2}{N}\in(0,\tfrac{1}{16})\) and pick a cutoff function \(\zeta=\zeta_r\in C^{\infty}\!\left(\mathcal{K}_r\right)\) satisfying \(\zeta_{\vert_{\partial \mathcal{K}_r}}=0\), \(0\leq \zeta\leq 1\) in \(\mathcal{K}_r\) as well as \(\zeta=1\) in \(\mathcal{K}_{(1-\lambda)r}\) and \(|\nabla\zeta|\leq\frac{4}{\lambda r}\). With these choices, we then take \(j=j(\Phi,M,q_1,q_2,a,\hat{b})\in\mathbb{N}\) so large that \[\begin{align} \frac{j^2}{(j-1)^2}\cdot\frac{1-2\gamma}{1-\gamma}&\leq 1-4\gamma^2,\\ \frac{j}{(j-1)^2\ln(2)}\cdot\frac{96C_2}{\lambda^2}&\leq \gamma^2,\\ C_1\theta_M^{-1}\omega_N^{-\frac{1}{q_2}}\Big(\frac{1+\ln(2)(M+1)j}{\ln^2(2)(j-1)^2}\Big)&\leq \gamma^2, \end{align}\] which is possible due to \(\gamma<\frac{1}{4}\), and let \(\delta_0=\delta_0(\Phi,M,q_1,q_2,a,\hat{b})=2^{-j}\in(0,\frac{1}{2})\). Moreover, we notice that the assumptions of the current lemma allow for an application of Lemma 9, yielding a time \(\tau\in\big[t_0-\theta r^2,t_0-\gamma\theta r^2\big]\) for which \[\begin{align} \label{eq:exp-in-time-eq1} \big|A^+_{\mu^+-k,r}(\tau)\big|\leq\frac{1-2\gamma}{1-\gamma}|\mathcal{K}_r|. \end{align}\tag{61}\] Having these preparations at hand, we now consider the cylinder \(Q:=Q_{r}(x_0,\tau,t)\) for arbitrary \(t\in(\tau,t_0)\). In view of \((\tau,t]\subseteq(t_0-\theta r^2,t_0]\) we have \(Q\subseteq Q_{r}^{\theta}\subseteq Q_{r}^{\vartheta}\) and hence also \(\mathop{\mathrm{ess\, sup}}_{Q} w\leq \mu^+\). Thus, we may draw on Lemma 6 for \(\xi=\xi_{\mu^+,k,\delta_0}\), as defined in 28 , inside the cylinder \(Q=Q_{r}(x_0,\tau,t)\) to find that \[\begin{align} \int_{\mathcal{K}_r}\!\!\xi^2\big(w(\cdot,t)\big)\zeta^2\leq\int_{\mathcal{K}_r}\!\!\xi^2\big(w(\cdot,\tau)\big)\zeta^2&+96\Phi'\big(\Phi^{-1}(\mu^+)\big)\frac{|t-\tau|}{\lambda^2 r^2}\ln\Big(\frac{1}{\delta_0}\Big)|\mathcal{K}_r|\\ &+C_1\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\Big(\int_{\tau}^t\big|A_{\mu^+-k,r}^+(s)\big|^{\frac{q_1(q_2-1)}{q_2(q_1-1)}}\mathop{\mathrm{d\!}}s\Big)^{\frac{q_1-1}{q_1}}, \end{align}\] where again we made use of the monotonicity of \(\Phi'\) and \(\Phi^{-1}\). Then, since \(t-\tau\leq t_0-t_0+\theta r^2=\theta r^2\) and \(\big|A^+_{\mu^+-k,r}\big|\leq |\mathcal{K}_r|\) for all \(s\in(\tau,t)\) we obtain \[\begin{align} \label{eq:exp-in-time-eq2} \int_{\mathcal{K}_r}\!\!\!\xi^2\big(w(\cdot,t)\big)\zeta^2\leq\!\int_{\mathcal{K}_r}\!\!\!\xi^2\big(w(\cdot,\tau)\big)\zeta^2&+\frac{96}{\lambda^2}\frac{\theta}{\theta_{\mu^+}}\ln\!\Big(\frac{1}{\delta_0}\Big)|\mathcal{K}_r|+C_1\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\big(\theta r^2\big)^\frac{q_1-1}{q_1}|\mathcal{K}_r|^{\frac{q_2-1}{q_2}} \end{align}\tag{62}\] for \(t\in(\tau,t_0)\). Now, if \(\mu^+-\mu^-\leq 4k\) and \(\mu^-\leq k\), we have \(\mu^+\leq 5k\) and in view of the subadditivity of \(\Phi^{-1}\) also \(\Phi^{-1}(\mu^+)\leq 5\Phi^{-1}(k)\). Similarly, if \(\mu^+-\mu^-\leq 4k\) and \(k<\mu^-\), we have \(\mu^+\leq 5\mu^-\) yielding \(\Phi^{-1}(\mu^+)\leq 5\Phi^{-1}(\mu^-)\), whence we find from 60 that in both cases \(\frac{\theta}{\theta_{\mu^+}}\leq C_2\). Therefore, since \(\xi(s)\equiv0\) for \(s\in[0,\mu^+-(1-\delta_0)k]\) with \(\xi(s)\leq\ln(\frac{1}{\delta_0})\) on \([0,\mu^+]\), we infer from a combination of 62 , 61 and 4 that \[\begin{align} \label{eq:exp-in-time-eq3} &\int_{\mathcal{K}_r}\!\!\!\xi^2\big(w(\cdot,t)\big)\zeta^2\nonumber\\ \leq\;&\ln^2\!\Big(\frac{1}{\delta_0}\Big)\big|A^+_{\mu^+-k,r}(\tau)\big|+96C_2\ln\!\Big(\frac{1}{\delta_0}\Big)\frac{|\mathcal{K}_r|}{\lambda^2}+C_1\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\big(\theta r^2\big)^\frac{q_1-1}{q_1}|\mathcal{K}_r|^{\frac{q_2-1}{q_2}}\nonumber\\ \leq\;&\ln^2\!\Big(\frac{1}{\delta_0}\Big)\frac{1-2\gamma}{1-\gamma}|\mathcal{K}_r|+96C_2\ln\Big(\frac{1}{\delta_0}\Big)\frac{|\mathcal{K}_r|}{\lambda^2}+C_1\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\theta^\frac{q_1-1}{q_1}\omega_N^{-\frac{1}{q_2}}r^{2-\frac{2}{q_1}-\frac{N}{q_2}}|\mathcal{K}_r|\nonumber\\ =\;&\left(\ln^2\!\Big(\frac{1}{\delta_0}\Big)\frac{1-2\gamma}{1-\gamma}+96C_2\ln\Big(\frac{1}{\delta_0}\Big)\frac{1}{\lambda^2}+C_1\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\theta^\frac{q_1-1}{q_1}\omega_N^{-\frac{1}{q_2}}r^{N\kappa}\right)|\mathcal{K}_r| \end{align}\tag{63}\] for \(t\in(\tau,t_0)\). Moreover, noticing that from the construction of \(\xi\) (see 28 ) we have \(\xi\big(w(x,t)\big)\geq \ln\big(\frac{1}{2\delta_0}\big)\) a.e. on \(\{w>\mu_+-\delta_0k\}\), we may estimate the left hand side from below against \[\begin{align} \label{eq:exp-in-time-eq4} \int_{\mathcal{K}_r}\xi^2\big(w(\cdot,t)\big)\zeta^2\geq \ln^2\Big(\frac{1}{2\delta_0}\Big)\big|\big\{x\in \mathcal{K}_{(1-\lambda)r}\,:\, w(x,t)>\mu^+-\delta_0 k\big\}\big|\quad\text{for }t\in(\tau,t_0), \end{align}\tag{64}\] because \(\zeta=1\) in \(\mathcal{K}_{(1-\lambda)r}\) and because \(\ln\big(\frac{1}{2\delta_0}\big)\) is positive due to the restriction \(\delta_0<\frac{1}{2}\). Next, we notice that by Bernoulli’s inequality we have \[\begin{align} \big|\mathcal{K}_{r}\setminus \mathcal{K}_{(1-\lambda)r}\big|=|\mathcal{K}_r|-|\mathcal{K}_{(1-\lambda)r}|=\big(1-(1-\lambda)^N\big)|\mathcal{K}_r|\leq N\lambda|\mathcal{K}_r| \end{align}\] and, accordingly, \[\begin{align} \big|A_{\mu^+-\delta_0 k,r}^+(t)\big|&=\big|\big\{x\in \mathcal{K}_{r}\,:\, w(x,t)>\mu^+-\delta_0 k\big\}\big|\\ &\leq\big|\big\{x\in \mathcal{K}_{(1-\lambda)r}\,:\, w(x,t)>\mu^+-\delta_0 k\big\}\big|+\big|\mathcal{K}_{r}\setminus \mathcal{K}_{(1-\lambda)r}\big|\\ &\leq\big|\big\{x\in \mathcal{K}_{(1-\lambda)r}\,:\, w(x,t)>\mu^+-\delta_0 k\big\}\big|+N\lambda|\mathcal{K}_r|. \end{align}\] Plugging 63 and 64 into this, we arrive at \[\begin{align} \label{eq:exp-in-time-eq5} &\big|A_{\mu^+-\delta_0 k,r}^+(t)\big|\\ \leq\;&\left(\frac{\ln^2(\frac{1}{\delta_0})(1-2\gamma)}{\ln^2(\frac{1}{2\delta_0})(1-\gamma)}+\frac{96C_2\ln(\frac{1}{\delta_0})}{\ln^2(\frac{1}{2\delta_0})\lambda^2}+\frac{C_1\theta_M^{-1}}{\ln^2(\frac{1}{2\delta_0})}\Big(\frac{1+\ln(\frac{1}{\delta_0})}{\delta_0^2 k^2}+\frac{\ln(\frac{1}{\delta_0})}{\delta_0 k}\Big)\theta^\frac{q_1-1}{q_1}\omega_N^{-\frac{1}{q_2}}r^{N\kappa}+N\lambda\right)|\mathcal{K}_r|\nonumber \end{align}\tag{65}\] for all \(t\in(\tau,t_0]\). Clearly, assuming \(\delta_0^2k^2\theta^{-\frac{q_1-1}{q_1}}>r^{N\kappa}\) to hold, we find that the facts that \(\tau \leq t_0-\tfrac{\varpi_0}{2}\theta r^2\) and \(k\leq M\) as well as our choices for \(\lambda,\delta_0\) and \(j\) finally entail \[\begin{align} \big|A_{\mu^+-\delta_0 k,r}^+(t)\big|\leq\big(1-4\gamma^2+\gamma^2+\gamma^2+\gamma^2\big)|\mathcal{K}_r|=\big(1-\gamma^2\big)|\mathcal{K}_r|=\Big(1-\frac{\varpi_0^2}{4}\Big)|\mathcal{K}_r| \end{align}\] for all \(t\in\big(t_0-\tfrac{\varpi_0}{2}\theta r^2,t_0\big]\) as claimed. ◻

Proof:. By definition, \(A^+_{\mu^+-\delta k,r}(t)\subseteq \mathcal{K}_r\) for all \(\delta\in(0,\delta_0)\) and \(t\in(0,T)\), so that \[\big|\mathcal{K}_r\setminus A^+_{\mu^+-\delta k,r}(t)\big|=|\mathcal{K}_r|-\big|A^+_{\mu^+-\delta k,r}(t)\big|\geq |\mathcal{K}_r|-\Big(1-\frac{\varpi_0^2}{4}\Big)|\mathcal{K}_r|=\frac{\varpi_0^2}{4}|\mathcal{K}_r|\] for all \(t\in\big(t_0-\tfrac{\varpi_0}{2}\theta r^2,t_0\big]\) is an evident consequence of Lemma 10. ◻

Proof:. With \(\delta_0\in(0,\frac{1}{2})\) provided by Lemma 10, we claim that the assertion holds if we take \(\delta_\star=\delta_0 2^{-j_\star}\) with sufficiently large \(j_\star=j_\star(\Phi,M,q_1,q_2,a,\hat{b},\varpi)\in\mathbb{N}\) to be determined later. For \(j\in\mathbb{N}_0\) we consider the numbers \(\delta_j:=\frac{\delta_0}{2^j}\). Obviously, \(\delta_j\leq \delta_0\) for all \(j\in\mathbb{N}_0\) and \(\delta_j-\delta_{j+1}=\delta_{j+1}=\frac{\delta_0}{2^{j+1}}\) for \(j\in\mathbb{N}_0\). To shorten the notation, we set \(\gamma:=\frac{\varpi_0}{2}\) and introduce \(A_j^+(t):=A_{\mu^+-\delta_j k,r}^+(t)\) for \(t\in(t_0-\gamma\theta r^2,t_0]\). Employing the De Giorgi inequality from Lemma 2, with \(L=\mu^+-\delta_{j+1} k\) and \(K=\mu^+-\delta_j k\), to the function \(x\mapsto w(x,t)\) entails that there is \(C_1>0\) such that \[\begin{align} \delta_{j+1}k\big|A_{j+1}^+(t)\big|=(\delta_j-\delta_{j+1})k\big|A_{j+1}^+(t)\big|\leq \frac{C_1r^{N+1}}{\big|\mathcal{K}_r\setminus A_j^+(t)\big|}\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big| \end{align}\] holds for all \(t\in(t_0-\gamma\theta r^2,t_0]\) and \(j\in\mathbb{N}_0\). The denominator on the right can be estimated using Corollary 4, so that integrating the resulting inequality over \((t_0-\gamma\theta r^2,t_0]\) and squaring yields \[\begin{align} \delta_{j+1}^2 k^2\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_{j+1}^+(t)\big|\Big)^2&\leq \Big(\frac{C_1r^{N+1}}{\gamma^2|\mathcal{K}_r|}\Big)^2\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big|\Big)^2\\ &=\frac{C_1^2r^2}{\gamma^4\omega_N^2}\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big|\Big)^2 \end{align}\] for all \(j\in\mathbb{N}_0\). Invoking Hölder’s inequality to split the last integral, we then arrive at the estimate \[\begin{align} \label{eq:exp-in-sp-eq1} &\delta_{j+1}^2 k^2 \Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_{j+1}^+(t)\big|\Big)^2\nonumber\\ \leq\;&\frac{C_1^2r^2}{\gamma^4\omega_N^2}\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big|^2\Big)\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A^+_j(t)\setminus A_{j+1}^+(t)\big|\Big) \end{align}\tag{67}\] for all \(j\in\mathbb{N}_0\). To control the gradient integral, we first note that with \(\psi\in C_0^\infty(\Omega_T)\) satisfying \(\psi\equiv 1\) on \(Q_{r}^{\gamma\theta}\), \(\psi\equiv 0\) on \(\Omega_T\setminus Q_{2r}^{\gamma\theta}\), \(|\nabla\psi|\leq\frac{4}{r}\) and \(\psi_t\leq \frac{2}{3\gamma\theta r^2}\) we obtain \[\begin{align} \int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big|^2&\leq \iint_{Q_{r}^{\gamma\theta}}\Big|\nabla\Big(\big(w(\cdot,t)-\mu^++\delta_j k\big)_+\psi\Big)\Big|^2\\ &\leq \iint_{Q_{2r}^{\gamma\theta}}\Big|\nabla\Big(\big(w(\cdot,t)-\mu^++\delta_j k\big)_+\psi\Big)\Big|^2\quad\text{for all }j\in\mathbb{N}_0, \end{align}\] so that drawing on Lemma 5 for \(\Lambda^+(w)=\Lambda^+(w,\mu^+-\delta_j k)\) in the cylinder \(Q^{\gamma\theta}_{2r}\) with \(\psi\) as described above provides \(C_2=C_2(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \label{eq:exp-in-sp-eq2} &\int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\big|\nabla w(\cdot,t)\big|^2\nonumber\\ \leq\;&2\iint_{Q_{2r}^{\gamma\theta}}\Lambda^+(w)\big(\psi^2\big)_t+3\iint_{Q_{2r}^{\gamma\theta}}(w-\mu^++\delta_j k)_+^2|\nabla\psi|^2\nonumber\\&+C_2\Big(\int_{t_0-4\gamma\theta r^2}^{t_0}\!\Big|\big\{x\in \mathcal{K}_{2r}:\big(w(x,t)-\mu^++\delta_j k\big)_+>0\big\}\Big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\tag{68}\] for all \(j\in\mathbb{N}_0\). Since \(\gamma\theta<\theta\leq\vartheta\) we have \(Q_{2r}^{\gamma\theta}\subseteq Q_{2r}^{\vartheta}\) and hence \(w\leq \mu^+\) in \(Q_{2r}^{\gamma\theta}\), so that \[\begin{align} (w-\mu^++\delta_j k)_+^2\leq \delta_j^2 k^2\quad\text{a.e. on }Q_{2r}^{\gamma\theta}. \end{align}\] Moreover noticing that by the assumption \(2k\leq \mu^+-\mu^-\) we have \(\mu^+-\delta_j k>\mu^+-k\geq \mu^-+k\), we may estimate \[\begin{align} \frac{\theta_{\mu^+-\delta_j k}}{\theta_k}=\frac{\Phi'\big(\Phi^{-1}(k)\big)}{\Phi'\big(\Phi^{-1}(\mu^+-\delta_j k)\big)}\leq \frac{\Phi'\big(\Phi^{-1}(k)\big)}{\Phi'\big(\Phi^{-1}(\mu^-+k)\big)}\leq 1\quad\text{and}\quad \frac{\theta_{\mu^+-\delta_j k}}{\theta_{\mu^-}}\leq\frac{\Phi'\big(\Phi^{-1}(\mu^-)\big)}{\Phi'\big(\Phi^{-1}(\mu^-+k)\big)}\leq 1, \end{align}\] which entails that in both cases for \(\theta\) we have \(\theta_{\mu^+-\delta_j k}\theta^{-1}\leq 1\). Thus, in view of the upper bound for \(\Lambda^+\) from Lemma 7 we obtain \[\begin{align} \frac{\Lambda^+(w)}{\theta}&\leq \frac{\theta_{\mu^+-\delta_j k}}{2\theta}(w-\mu^++\delta_j k)_+^2\leq (w-\mu^++\delta_j k)_+^2\leq \delta_j^2 k^2\quad\text{a.e. on }Q_{2r}^{\gamma\theta}. \end{align}\] Making use of these estimates, the properties of \(\psi\) and 4 , we can further refine 68 to \[\begin{align} \label{eq:exp-in-sp-eq3} \int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\!\!\big|\nabla w(\cdot,t)\big|^2&\leq\frac{8\delta_j^2 k^2}{3\gamma r^2}\big|Q_{2r}^{\gamma\theta}\big| +\frac{48\delta_j^2 k^2}{r^2}\big|Q_{2r}^{\gamma\theta}\big|+C_2|\mathcal{K}_{2r}|^{\frac{q_2-1}{q_2}}\big|4\gamma\theta r^2\big|^\frac{q_1-1}{q_1}\nonumber\\ &=\frac{\delta_j^2 k^2}{r^2}\Big(\frac{8}{3\gamma}+48\Big)\big|Q_{2r}^{\gamma\theta}\big|+C_2 2^{N+2-\frac{N}{q_2}-\frac{2}{q_1}}\omega_N^{1-\frac{1}{q_2}}r^{N+N\kappa}\gamma^{1-\frac{1}{q_1}}\theta^{\frac{q_1-1}{q_1}}\nonumber\\ &=\frac{\delta_j^2 k^2}{r^2}\Big(\frac{8}{3\gamma}+48+\frac{C_2r^{N\kappa}\theta^{\frac{q_1-1}{q_1}}}{\delta_j^2 k^2}2^{N\kappa-2}\omega_N^{-\frac{1}{q_2}}\theta^{-2}\gamma^{-\frac{1}{q_1}}\Big)\big|Q_{2r}^{\gamma\theta}\big| \end{align}\tag{69}\] for all \(j\in\mathbb{N}_0\). Now, assuming 66 to be false we have \(r^{N\kappa}<\delta_\star^2 k^2\theta^{-\frac{q_1-1}{q_1}}\), so that we conclude from the fact that due to the monotonicity of \(\Phi'\) and \(\Phi^{-1}\) and \(\mu^-\leq M\) and \(k\leq M\) we have \(\theta^{-2}\leq\theta_M^{-2}\) and 69 , that there is \(C_3=C_3(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \int_{t_0-\gamma\theta r^2}^{t_0}\!\int_{A_j^+(t)\setminus A_{j+1}^+(t)}\!\!\big|\nabla w(\cdot,t)\big|^2&\leq C_3\frac{\delta_j^2 k^2}{r^2}\Big(1+\frac{\delta_\star^2}{\delta_j^2}\Big)\big|Q_{2r}^{\gamma\theta}\big| \end{align}\] for all \(j\in\mathbb{N}_0\). Plugging this back into 67 and reordering yields \[\begin{align} \Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_{j+1}^+(t)\big|\Big)^2\leq \frac{C_1^2r^2}{\gamma^4\omega_N^2}\cdot C_3\frac{\delta_j^2 k^2}{\delta_{j+1}^2 k^2 r^2}\Big(1+\frac{\delta_\star^2}{\delta_j^2}\Big)\big|Q_{2r}^{\gamma\theta}\big|\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_j^+(t)\setminus A_{j+1}^+(t)\big|\Big) \end{align}\] for all \(j\in\mathbb{N}_0\). Then, since by definition we have \(\delta_{j}\geq\delta_\star\) for all \(j\leq j_\star\) and \(\frac{\delta_j}{\delta_{j+1}}=2\) for all \(j\leq j_\star-1\), we find that there is \(C_4=C_4(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \label{eq:exp-in-sp-eq4} \Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_{j+1}^+(t)\big|\Big)^2\leq C_4\big|Q_{2r}^{\gamma\theta}\big|\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_j^+(t)\setminus A_{j+1}^+(t)\big|\Big)\quad\text{for all }0\leq j\leq j_\star -1 \end{align}\tag{70}\] and a summation of 70 from \(j=0\) to \(j=j_\star-1\) entails \[\begin{align} &j_\star\big|\big\{(x,t)\in Q_{r}^{\gamma\theta}\,:\, (w-\mu^++\delta_\star k)_+>0\big\}\big|^2\\ =\;& j_\star\Big(\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_{j_\star}^+(t)\big|\Big)^2\\ \leq\;&C_4\big|Q_{2r}^{\gamma\theta}\big|\cdot\sum_{j=0}^{j_\star-1}\int_{t_0-\gamma\theta r^2}^{t_0}\big|A_j^+(t)\setminus A_{j+1}^+(t)\big|\\ \leq\;&C_4|Q_{2r}^{\gamma\theta}\big|\cdot\big|\big\{(x,t)\in Q_{r}^{\gamma\theta}\,:\, (w-\mu^++\delta_0 k)_+>0\big\}\big|\\ \leq\;&C_4|Q_{2r}^{\gamma\theta}\big|\cdot\big|Q_{r}^{\gamma\theta}\big| \end{align}\] in view of the nested inclusions \(A_{j+1}^+\subseteq A_j^+(t)\) for all \(j\leq j_\star-1\) and \(t\in(t_0-\gamma\theta r^2,t_0]\). Rewriting \(|Q_{2r}^{\gamma\theta}|=2^{N+2}|Q_{r}^{\gamma\theta}|\) and dividing by \(j_\star\) we obtain \[\begin{align} \big|\big\{(x,t)\in Q_{r}^{\gamma\theta}:(w-\mu^++\delta_\star k)_+>0\big\}\big|^2\leq \frac{C_4 2^{N+2}}{j_\star}|Q_{r}^{\gamma\theta}|^2. \end{align}\] Accordingly, our claim is proven if we take \(j_\star=j_\star(\Phi,M,q_1,q_2,a,\hat{b},\varpi)\in\mathbb{N}\) so large that \(\sqrt{\frac{C_4 2^{N+2}}{j_\star}}\leq \varpi.\) ◻

Proof:. We repeat the iteration procedure of Lemma 8 in an adjusted manner. (See also [42].) We claim that the assertion holds if we pick \(\varpi_1\in(0,1)\) sufficiently small with our choice only depending on \(\Phi,M,a\) and \(\hat{b}\), and then let \(\delta_\diamond=\delta_\diamond(\Phi,M,q_1,q_2,a,\hat{b})=\delta_\star(\Phi,M,q_1,q_2,a,\hat{b},\varpi_1)\!\in(0,\frac{1}{4})\), where \(\delta_\star\) is the number provided by Lemma 11. We set \(\gamma:=\frac{\varpi_0}{2}\), \(k_0:=\mu^+-\delta_\diamond k\), \(r_0=r\) and define \[\begin{align} k_i:=\mu^+-\frac{\delta_\diamond k}{2}-\frac{\delta_\diamond k}{2^{i+1}},\quad\text{and}\quad r_i=\frac{r}{2}+\frac{r}{2^{i+1}}. \end{align}\] Accordingly, for all \(i\in\mathbb{N}_0\) we have \(k_i\in[\mu^+-\delta_\diamond k,\mu^+-\frac{\delta_\diamond k}{2})\), \(r_i\in(\frac{r}{2},r]\), \(k_i\leq k_{i+1}\) and, moreover, \(r_\infty:=\lim_{i\to\infty} r_i=\frac{r}{2}\) and \(k_\infty:=\lim_{i\to\infty} k_i=\mu^+-\frac{\delta_\diamond k}{2}\). Additionally, we see that the assumption \(\mu^+-\mu^-\geq 2k\) implies that \(k_0>\mu^+-k\geq \mu^-+k\). For \(i\in\mathbb{N}_0\) we denote \(Q_i:=Q_{r_i}^{\gamma\theta}=\mathcal{K}_{r_i}(x_0)\times(t_0-\gamma\theta r_i^2,t_0]\) and fix cutoff functions similar to those in Lemma 8. In fact, we take \(\psi_i\in C_0^\infty\big(\Omega_T\big)\) with \(\mathop{\mathrm{supp}}\psi_i\subseteq \overline{Q_i}\) such that \[\begin{align} \psi_i\equiv 1\;\text{in}\;Q_{i+1},\qquad \psi_i\equiv 0\text{ in }\partial_p Q_i,\qquad |\nabla\psi_i|\leq \frac{2^{i+3}}{r}\qquad\text{and}\qquad |(\psi_i)_t|\leq\frac{2^{2(i+2)}}{\gamma\theta r^2}. \end{align}\] This way, \(Q_i\subseteq Q_{r_i}^{\theta}\subseteq Q_{2r}^{\vartheta}\) for all \(i\in\mathbb{N}_0\) and hence \(w\leq \mu^+\) in \(Q_i\) for all \(i\in\mathbb{N}_0\). We proceed by employing Lemma 5 for \(\Lambda^+(w)=\Lambda^+(w,k_i)\) in \(Q_i\) and obtain \(\Gamma_1=\Gamma_1(\Phi,M,q_1,q_2,a,\hat{b})>0\) satisfying \[\begin{align} &\sup_{t_0-\gamma\theta r_i^2\leq t \leq t_0}2\int_{\mathcal{K}_{r_i}}\Lambda^+\big(w(\cdot,t)\big)\psi_i^2(\cdot,t)+\iint_{Q_i}\big|\nabla\big((w-k_i)_+\psi_i\big)\big|^2\\ \leq\;&4\iint_{Q_{i}}\Lambda^+(w)\psi_i|(\psi_i)_t|+3\iint_{Q_i}(w-k_i)_+^2|\nabla\psi_i|^2+\Gamma_1\Big(\int_{t_0-\gamma\theta r_i^2}^{t_0}\!\big|A_{k_i,r_i}^+(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] for all \(i\in\mathbb{N}_0\). Recalling that in view of Lemma 7 we have \[\begin{align} \frac{\theta_{\mu^+}}{2}(w-k_i)_+^2=\frac{(w-k_i)_+^2}{2\Phi'\big(\Phi^{-1}(\mu^+)\big)}\leq \Lambda^+(w)\leq \frac{(w-k_i)_+^2}{2\Phi'\big(\Phi^{-1}(k_i)\big)}=\frac{\theta_{k_i}}{2}(w-k_i)_-^2\quad\text{in }Q_i \end{align}\] yields \[\begin{align} &J_i:=\sup_{t_0-\gamma\theta r_i^2\leq t \leq t_0}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,t)-k_i)_+^2\psi_i^2(\cdot,t)+\frac{1}{\theta_{\mu^+}}\iint_{Q_i}\big|\nabla\big((w-k_i)_+\psi_i\big)\big|^2\nonumber\\ &\qquad\leq \frac{2^{2i+5}}{\gamma r^2}\frac{\theta_{k_i}}{\theta_{\mu^+}\theta}\iint_{Q_{i}}(w-k_i)_+^2+\frac{2^{2i+8}}{\theta_{\mu^+}r^2}\iint_{Q_i}(w-k_i)_+^2+\frac{\Gamma_1}{\theta_{\mu^+}}\Big(\int_{t_0-\gamma\theta r_i^2}^{t_0}\!\big|A_{k_i,r_i}^+(t)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] for all \(i\in\mathbb{N}_0\) in view of the monotonicity of \(\Phi'\), \(\Phi^{-1}\) and the properties of \(\psi_i\). As \(\frac{\theta}{\theta_{\mu^+}}\geq 1\) due to \(\mu^+\geq k\) and \(\mu^+\geq \mu^-\), we obtain from repeating the change of variables \(\bar{t}=\frac{t-t_0}{\theta}\) as in Lemma 8 and denoting \(\bar{Q}_i:=\mathcal{K}_{r_i}\times(-\gamma r_i^2,0]\) and \(\bar{A}^+_i(\bar{t}):=A_{k_i,r_i}^+(\theta\bar{t}+t_0)\) that \[\begin{align} \label{eq:sec-alt-eq1} &\bar{J}_i:=\sup_{-\gamma r_i^2\leq \bar{t} \leq 0}\int_{\mathcal{K}_{r_i}}\big(w(\cdot,\bar{t})-k_i)_+^2\psi_i^2(\cdot,\bar{t})+\iint_{\bar{Q}_i}\big|\nabla\big((w-k_i)_+\psi_i\big)\big|^2\nonumber\\ &\qquad\leq \frac{2^{2i+5}}{\gamma r^2}\frac{\theta_{k_i}}{\theta_{\mu^+}}\iint_{\bar{Q}_{i}}(w-k_i)_+^2+\frac{2^{2i+8}}{r^2}\frac{\theta}{\theta_{\mu^+}}\iint_{\bar{Q}_i}(w-k_i)_+^2+\Gamma_1\frac{\theta^{\frac{q_1-1}{q_1}}}{\theta\theta_{\mu^+}}\Big(\int_{-\gamma r_i^2}^{0}\!\big|\bar{A}_{i}^+(\bar{t})\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\mathop{\mathrm{d\!}}\bar{t}\Big)^\frac{q_1-1}{q_1} \end{align}\tag{73}\] for all \(i\in\mathbb{N}_0\). Here, we note that our choice for \(k_i\) and the assumption \(\mu^+-\mu^-\geq 2k\) entail that \[\begin{align} k_i\geq k_0=\mu^+-\delta_\diamond k> \mu^+- k\geq\mu^-+k\geq\max\{\mu^-,k\}\quad\text{for all }i\in\mathbb{N}_0, \end{align}\] so that in both instances for \(\theta\) we have \(\theta_{k_i}\leq \theta\) due to the monotonicity properties of \(\Phi'\) and \(\Phi^{-1}\). Moreover, the condition \(\mu^+-\mu^-\leq 4k\) entails that \(\mu^+\leq 5\max\{k,\mu^-\}\leq 5M\), so that in either case we may conclude from Lemma 1, again utilizing the monotonicity of \(\Phi'\) and \(\Phi^{-1}\) as well as the subadditivity of \(\Phi^{-1}\), that there is \(C_1=C_1(\Phi,M)>0\) such that \[\begin{align} \frac{\theta_{k_i}}{\theta_{\mu^+}}\leq\frac{\theta}{\theta_{\mu^+}}\leq \frac{\Phi'\big(\Phi^{-1}(5\max\{k,\mu^-\})\big)}{\Phi'\big(\Phi^{-1}(\max\{k,\mu^-\})\big)}\leq \frac{\Phi'\big(5\Phi^{-1}(\max\{k,\mu^-\})\big)}{\Phi'\big(\Phi^{-1}(\max\{k,\mu^-\})\big)}\leq C_1\quad\text{for all }i\in\mathbb{N}_0. \end{align}\] Plugging this back into 73 , estimating \(\theta^{-1}\leq\theta_{\mu^+}^{-1}\leq\theta_M^{-1}\) as well as \((w-k_i)_+\leq \delta_\diamond k\) for all \(i\in\mathbb{N}_0\) and introducing the notation \(\bar{A}_i^+:=\bar{Q}_i\cap\big\{(x,t)\in\Omega\times(-\frac{t_0}{\theta},\frac{T-t_0}{\theta}):\big(w(x,\bar{t})-k_i\big)_+>0\big\}\), we find \(\Gamma_2=\Gamma_2(\Phi,M,q_1,q_2,a,\hat{b})>0\) fulfilling \[\begin{align} \label{eq:sec-alt-eq2} \bar{J}_i\leq \Gamma_2\frac{2^{2i}}{r^2}\Big(\frac{1}{\gamma}+1\Big)\delta_\diamond^2k^2\big|\bar{A}_i^+\big|+\Gamma_2\theta^{\frac{q_1-1}{q_1}}\Big(\int_{-\gamma r_i^2}^{0}\big|\bar{A}_{i}^+(\bar{t})\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\mathop{\mathrm{d\!}}{\bar{t}}\Big)^\frac{q_1-1}{q_1} \end{align}\tag{74}\] for all \(i\in\mathbb{N}_0\). Repeating steps similar to those in Lemma 8, while drawing on Corollary 3 and the fact that \((w-k_i)_+\geq k_{i+1}-k_i=\frac{\delta_\diamond k}{2^{i+2}}\) a.e. in \(\bar{Q}_{i+1}\), we moreover see that there is \(C_2>0\) such that \[\begin{align} \label{eq:sec-alt-eq3} \big|\bar{A}_{i+1}^+\big|\leq \frac{C_2 2^{2i+4}}{\delta_\diamond^2 k^2}\big|\bar{A}_i^+\big|^\frac{2}{N+2}\bar{J}_i\quad\text{for }i\in\mathbb{N}_0. \end{align}\tag{75}\] Setting \(\widehat{q}_j:=\frac{2q_j(1+\kappa)}{q_j-1}\) for \(j\in\{1,2\}\) and \(\kappa\) as in 4 as well as \[\begin{align} \bar{Y}_i:=\frac{\big|\bar{A}_i^+\big|}{r^{N+2}}\quad\text{as well as}\quad\bar{Z}_i:=\frac{1}{r^N}\Big(\int_{-\gamma r_i^2}^0\big|\bar{A}_{i}^+(\bar{t})\big|^{\frac{\widehat{q_1}}{\widehat{q_2}}}\mathop{\mathrm{d\!}}{\bar{t}}\Big)^{\frac{2}{\widehat{q_1}}},\quad i\in\mathbb{N}_0, \end{align}\] we conclude from 74 and 75 that there is \(\Gamma_3=\Gamma_3(\Phi,M,q_1,q_2,a,\hat{b})>0\) satisfying \[\begin{align} \bar{Y}_{i+1}\leq\Gamma_3 2^{4i}\Big(\frac{1}{\gamma}+1\Big)\bar{Y}_i^{1+\frac{2}{N+2}}+\Gamma_3 2^{4i}\frac{r^{N\kappa}\theta^{\frac{q_1-1}{q_1}}}{\delta_\diamond^2 k^2}\bar{Y}_i^{\frac{2}{N+2}}\bar{Z}_{i}^{1+\kappa}\quad\text{for all }i\in\mathbb{N}_0. \end{align}\] As the choice for \(\gamma=\frac{\varpi_0}{2}\) only depends on \(\Phi,M,q_1,q_2,a\) and \(\hat{b}\), we find that if 71 fails there is \(\Gamma_4=\Gamma_4(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \bar{Y}_{i+1}\leq\Gamma_4 2^{4i}\Big(\bar{Y}_i^{1+\frac{2}{N+2}}+\bar{Y}_i^{\frac{2}{N+2}}\bar{Z}_i^{1+\kappa}\Big) \end{align}\] Mirroring the steps from Lemma 8 for \(\bar{Z}_i\) we also infer from Lemma 3 that if 71 fails there are \(C_3>0\) and \(\Gamma_5=\Gamma_5(\Phi,M,q_1,q_2,a,\hat{b})>0\) such that \[\begin{align} \bar{Z}_{i+1}\leq\frac{2^{2i+4}C_3}{r^N\delta_\diamond^2 k^2}\bar{J}_i\leq 2^{4i}\Gamma_5 \bar{Y}_i+2^{4i}\Gamma_5\frac{r^{N\kappa}\theta^{\frac{q_1-1}{q_1}}}{\delta_\diamond^2 k^2}\bar{Z}_i^{1+\kappa}\leq 2^{4i}\Gamma_5\Big(\bar{Y}_i+\bar{Z}_i^{1+\kappa}\Big)\quad\text{for all }i\in\mathbb{N}_0, \end{align}\] so that once more we are left with checking the condition on \(\bar{Y}_0+\bar{Z}_0^{1+\kappa}\) from Lemma 4. One can easily check (compare the steps leading to 56 and 57 ) that according to the change in variable and 4 we have \[\begin{align} \label{eq:sec-alt-eq4} \bar{Y}_0+\bar{Z}_0^{1+\kappa}&\leq \frac{\big|\bar{A}_{0}^+\big|}{r^{N+2}}+\omega_N^{\max\{\frac{1}{q_1}-\frac{1}{q_2},0\}}r^{\frac{N+2}{\min\{q_1,q_2\}}-N-2}\big|\bar{A}_{0}^+\big|^{q}\nonumber\\&\leq\frac{\big|A_{0}^+\big|}{\theta r^{N+2}}+\omega_N^{\max\{\frac{1}{q_1}-\frac{1}{q_2},0\}}r^{\frac{N+2}{\min\{q_1,q_2\}}-N-2}\theta^{-q}\big|A_{0}^+\big|^q, \end{align}\tag{76}\] where we wrote \(q=\min\big\{\frac{q_1-1}{q_1},\frac{q_2-2}{q_2}\big\}\) and \(A_0^+=\big\{(x,t)\in Q_r^{\gamma\theta}:(w-\mu^++\delta_\diamond k)_+>0\}\). Since we assume that 71 fails, we find from our definition of \(\delta_\diamond=\delta_\star\) and Lemma 11 that \[\begin{align} \big|A_{0}^+\big|=\Big|\Big\{(x,t)\in Q_{r}^{\gamma\theta}:(w-\mu^++\delta_\diamond k)_+>0\Big\}\Big|\leq \varpi_1\big|Q_{r}^{\gamma\theta}\big|=\varpi_1\gamma\theta\omega_Nr^{N+2}, \end{align}\] so that 76 turns into \[\begin{align} \bar{Y}_0+\bar{Z}_0^{1+\kappa}\leq \varpi_1\gamma\omega_N+\big(\varpi_1\gamma\omega_N\big)^{\frac{q_2-1}{q_2}}r^{2-\frac{2}{q_2}}\leq\varpi_1\gamma\omega_N+\big(\varpi_1\gamma\omega_N\big)^{\frac{q_2-1}{q_2}} \end{align}\] because of \(r<1\) and \(q_2>1\). As \(\gamma=\frac{\varpi_0}{2}\) and \(\varpi_0\) only depends \(\Phi,M,q_1,q_2,a,\hat{b}\), we can therefore pick \(\varpi_1=\varpi_1(\Phi,M,q_1,q_2,a,\hat{b})>0\) so small that with \(\sigma=\min\{\kappa,\frac{2}{N+2}\}\) and \(\Gamma_6=\Gamma_6(\Phi,M,q_1,q_2,a,\hat{b}):=\max\{\Gamma_4,\Gamma_5\}\) we have \[\begin{align} \varpi_1\gamma\omega_N+\big(\varpi_1\gamma\omega_N\big)^{\frac{q_2-1}{q_2}}\leq \big(2\Gamma_6\big)^{-\frac{1+\kappa}{\sigma}}2^{-\frac{4(1+\kappa)}{\sigma^2}}, \end{align}\] thereby making Lemma 4 applicable to say \(\bar{Y}_i\to 0\) as \(i\to\infty\). This immediately yields \[\begin{align} \big|\big\{(x,t)\in Q_{\frac{r}{2}}^{\gamma\theta}:(w-\mu^++\tfrac{\delta_\diamond}{2}k)_+>0\big\}\big|&=\theta\big|\big\{(x,\bar{t})\in \bar{Q}_{\infty}:(w-\mu^++\tfrac{\delta_\diamond}{2}k)_+>0\big\}\big|=\theta\big|\bar{A}^+_\infty\big|=0 \end{align}\] and hence 72 . ◻

Proof:. With \(r_0\), \(k_0\) and \(\theta_0:=\theta_{k_0}=\big(\Phi'(\Phi^{-1}(k_0))\big)^{-1}\) provided we set \(Q_0:=Q_{r_0}^{\theta_{0}}\), \(\mu^-_{0}:=\mathop{\mathrm{ess\,inf}}_{Q_0} w\) and \(\mu^+_{0}:=\mathop{\mathrm{ess\, sup}}_{Q_0}w\). If \(\mu^-_{0}>k_0\) we set \(j_\star=j_0\) and have nothing to prove. Assume now \(\mu^-_{0}\leq k_0\). For \(j>0\) we denote by \(\mu^-_{j}:=\mathop{\mathrm{ess\,inf}}_{Q_j} w\) and \(\mu^+_{j}:=\mathop{\mathrm{ess\, sup}}_{Q_j}w\) the infimum and supremum, respectively, in the current cylinder \(Q_j\). If at some point \(\mu^-_{j}>k_j\), we are finished. If not, we will generate \(r_{j+1}\) and \(k_{j+1}\) depending on which of the different cases we find ourselves in. The next scaling factor \(\theta_{j+1}\) will always be chosen in dependence of \(k_{j+1}\), i.e. \(\theta_{j+1}:=\theta_{k_{j+1}}\) in all cases, so that in fact \(\theta_j=\theta_{k_j}\) for all \(j\leq j_\star\). Before giving details for the different cases, however, let us fix \(C_1=C_1(\Phi,2,M)>1\) such that \[\begin{align} \label{eq:it1-c1} \Phi'\big(\Phi^{-1}(s)\big)=\Phi'\big(2\cdot\tfrac12\Phi^{-1}(s)\big)\leq C_1\Phi'\big(\tfrac12\Phi^{-1}(s)\big)\quad\text{for all }s\in[0,M], \end{align}\tag{78}\] which is possible in view of Lemma 1. Moreover, let the numbers \(\varpi_0=\varpi_0(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac12)\) and \(\delta_\diamond=\delta_\diamond(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac14)\) be given by Lemma 8 and Lemma 12, respectively. We claim that the lemma holds for the choices \[\label{eq:it1-const} \begin{align} \lambda&:=\frac{\sqrt{\varpi_0}}{4\sqrt{2C_1}}\in(0,\tfrac18),\qquad\eta:=1-\tfrac{\delta_\diamond}{8}\in(\tfrac{31}{32},1),\qquad \beta:=\frac{N\kappa}{2+\frac{q_1-1}{q_1}}<1\\\text{ and } \; C&:=\max\Big\{1,\Big(\big(\Phi^{-1}(M)\big)^{\frac{q_1-1}{q_1}}\delta_\diamond^{-2}\Big)^\frac{1}{2+\frac{q_1-1}{q_1}}\Big\}\geq 1. \end{align}\tag{79}\] Let us now turn to the construction of the next cylinder.

Case 1: \(\mathop{\mathrm{ess\,osc}}_{Q_j} w< 2k_j\).
With \(C_1>1\) as provided above, we set \(r_{j+1}=\frac{r_j}{2\sqrt{C_1}}\), \(k_{j+1}=\frac{k_j}{2}\) and \(\theta_{j+1}:=\theta_{k_{j+1}}\). Clearly, the conditions for \(k_{j+1}\) and \(r_{j+1}\) in 77 are satisfied in view of \(\eta>\tfrac12\) and \(\lambda<\tfrac1{2\sqrt{C_1}}\). Concerning the first condition, we note that 78 , the monotonicity of \(\Phi'\) and the subadditivity of \(\Phi^{-1}\) imply that \[\Phi'\big(\Phi^{-1}(k_j)\big)\leq C_1\Phi'\big(\tfrac{1}{2}\Phi^{-1}(k_j)\big)=C_1\Phi'\big(\tfrac{1}{2}\Phi^{-1}(2\cdot\tfrac{k_j}{2})\big)\leq C_1\Phi'\big(\Phi^{-1}(\tfrac{k_j}{2})\big).\] Thus, we have \[\begin{align} r_{j+1}^2\theta_{{j+1}}=\frac{r_j^2}{4C_1\Phi'\big(\Phi^{-1}(\frac{k_j}{2})\big)}\leq \frac{r_j^2}{4\Phi'\big(\Phi^{-1}(k_j)\big)}=\frac{r_j^2}{4}\theta_{k_j}\leq r_j^2\theta_{j}, \end{align}\] so that \(Q_{j+1}\subseteq Q_{j}\) follows from our choice \(r_{j+1}\leq \frac{r_j}{2}\). We are left with verifying \(\mathop{\mathrm{ess\,osc}}_{Q_{j+1}}w\leq 4 k_{j+1}\). This is an evident consequence of the nesting of cylinders, our case assumption and the particular choice of \(k_{j+1}\). In fact, due to \(Q_{j+1}\subseteq Q_j\) we have \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w<2k_j=4 k_{j+1}. \end{align}\]

Case 2: \(2k_j\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4k_j\).
We further distinguish cases indicated by the alternatives in Lemma 8 and Lemma 12, which will be employed for \(\vartheta=\theta_j\), \(r=\frac{r_j}{2}\) and \(k=k_j\). Note that due to the standing assumption that \(k_j\geq \mu^-_{j}\) we have \(\theta=\theta_{k_j}\) in these lemmas.

Case 2a: \(\delta_\diamond^2 k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}\leq \big(\tfrac{r_j}{2}\big)^{N\kappa}\), i.e. 71 is true for \(k=k_j\), \(\theta=\theta_{k_j}\) and \(r=\tfrac{r_j}{2}\).
In this case we set \(r_{j+1}:=\frac{r_j}{2}\), \(k_{j+1}:=k_j\) and \(\theta_{j+1}:=\theta_{k_{j+1}}\). These choices entail that \[\begin{align} r_{j+1}^2\theta_{k_{j+1}}=\frac{r_j^2}{4}\theta_{k_j}\leq r_j^2\theta_{k_j}\quad\text{and}\quad r_{j+1}\leq r_{j}, \end{align}\] and hence \(Q_{j+1}\subseteq Q_{j}\). Due to this and the choice \(k_{j+1}=k_j\) we also immediately obtain \[\mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq\mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4{k_j}=4 k_{j+1}.\] Fulfillment of \(\frac{1}{2} k_{j}\leq k_{j+1}\) and \(\lambda r_{j}\leq r_{j+1}\leq \frac{1}{2}r_{j}\) are also evident by the given choices due to \(\lambda<\frac{1}{2}\). We are left with checking the upper bound for \(k_{j+1}\). In view of the mean value theorem, the monotonicity of \(\Phi'\) and \(\Phi(0)=0\) we find that \(\Phi(s)\leq s\Phi'(s)\) for all \(s>0\), so that with \(s=\Phi^{-1}(k_j)>0\) we get \[\begin{align} k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}\big(\Phi^{-1}(k_j)\big)^{\frac{q_1-1}{q_1}}=k_j^2\big(\Phi'\big(\Phi^{-1}(k_j)\big)\big)^{\frac{q_1-1}{q_1}}\big(\Phi^{-1}(k_j)\big)^{\frac{q_1-1}{q_1}}\geq k_j^{2+\frac{q_1-1}{q_1}}. \end{align}\] In particular, utilizing 71 and monotonicity of \(\Phi^{-1}\) we have \[\begin{align} \delta_\diamond^2 k_j^{2+\frac{q_1-1}{q_1}}<\big(\Phi^{-1}(k_j)\big)^{\frac{q_1-1}{q_1}}\Big(\frac{r_j}{2}\Big)^{N\kappa}\leq \big(\Phi^{-1}(M)\big)^{\frac{q_1-1}{q_1}}r_j^{N\kappa}. \end{align}\] Recalling that we took \(C=\max\Big\{1,\Big(\big(\Phi^{-1}(M)\big)^{\frac{q_1-1}{q_1}}\delta_\diamond^{-2}\Big)^\frac{1}{2+\frac{q_1-1}{q_1}}\Big\}\geq 1\) and \(\beta=\frac{N\kappa}{2+\frac{q_1-1}{q_1}}<1\), we therefore obtain \[\begin{align} k_{j+1}=k_j\leq \Big(\big(\Phi^{-1}(M)\big)^{\frac{q_1-1}{q_1}}\delta_\diamond^{-2}r_j^{N\kappa}\Big)^\frac{1}{2+\frac{q_1-1}{q_1}}\leq Cr_j^\beta. \end{align}\]

Case 2b: \(\delta_\diamond^2 k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}> \big(\tfrac{r_j}{2}\big)^{N\kappa}\) and \(\big|\big\{(x,t)\in Q_{\frac{r_j}{2}}^{\theta_{k_j}}:(w-\mu^-_{j}-k_j)_->0\big\}\big|\leq \varpi_0\big|Q_{\frac{r_j}{2}}^{\theta_{k_j}}\big|\), i.e 71 is false and 45 is true for \(r=\frac{r_j}{2}\), \(k=k_j\) and \(\theta=\theta_{k_j}\).
Note that \(\delta_\diamond^2 k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}>\big(\frac{r_j}{2}\big)^{N\kappa}\) also entails that 46 is false, since \(\delta_\diamond<\frac{1}{4}\). Accordingly, we conclude from Lemma 8 that \[\begin{align} \Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\theta_{k_j}}:\big(w-\mu^-_{j}-\tfrac{k_j}{2}\big)_->0\Big\}\Big|=0, \end{align}\] implying that \[\begin{align} \label{eq:it1-2b-inf} {\textstyle\mathop{\mathrm{ess\,inf}}_{Q_{\frac{r_j}{4}}^{\theta_{k_j}}}} w>\mu^-_{j}+\frac{k_j}{2}. \end{align}\tag{80}\] Similar to Case 1, we find from 78 , monotonicity of \(\Phi'\) and subadditivity of \(\Phi^{-1}\) that \[\begin{align} \Phi'\big(\Phi^{-1}(k_j)\big)\leq C_1\Phi'\big(\tfrac12\Phi^{-1}(k_j)\big)< C_1\Phi'\big(\tfrac12\Phi^{-1}(\tfrac74 k_j)\big)\leq C_1\Phi'\big(\Phi^{-1}(\tfrac78 k_j)\big). \end{align}\] Thus, choosing \(r_{j+1}:=\frac{r_j}{4\sqrt{C_1}}\), \(k_{j+1}:=\frac{7}{8}k_j\) and \(\theta_{j+1}:=\theta_{k_{j+1}}\) with \(C_1>1\) as in 78 , we obtain that \[\begin{align} r_{j+1}^2\theta_{k_{j+1}}=\frac{r_j^2}{16 C_1\Phi'\big(\Phi^{-1}(\tfrac78 k_j)\big)}\leq\frac{r_j^2}{16\Phi'\big(\Phi^{-1}(k_j)\big)}\leq \Big(\frac{r_j}{4}\Big)^2\theta_{k_j}\quad\text{and}\quad r_{j+1}\leq\frac{r_j}{4}, \end{align}\] so that in fact \(Q_{j+1}\subseteq Q_{\frac{r_j}{4}}^{\theta_{k_j}}\subseteq Q_j\). Hence, we may use 80 and \(\mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4 k_{j}\) to estimate \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \textstyle{\mathop{\mathrm{ess\,osc}}_{Q_{\frac{r_j}{4}}^{\theta_{k_j}}}} w\leq \mu^+_{j}-\mu^-_{j}-\frac{k_j}{2}\leq 4k_j-\frac{k_j}{2}=\frac{7}{2}k_j=4 k_{j+1}. \end{align}\] Since the properties for \(k_{j+1}\) and \(r_{j+1}\) are clearly satisfied because \(\eta>\frac{31}{32}\) and \(\lambda<\frac{1}{4\sqrt{C_1}}\), we are finished with this case.

Case 2c: \(\delta_\diamond^2 k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}> \big(\tfrac{r_j}{2}\big)^{N\kappa}\) and \(\big|\big\{(x,t)\in Q_{\frac{r_j}{2}}^{\theta_{k_j}}:(w-\mu^+_{j}+k_j)_+>0\big\}\big|\leq (1-\varpi_0)\big|Q_{\frac{r_j}{2}}^{\theta_{k_j}}\big|\), i.e. 71 and 45 are both false. (Recall the reasoning at the start of Section 4.2.)
Here, writing \(\gamma:=\frac{\varpi_0}{2}<\frac{1}{4}\), we can draw on Lemma 12 to obtain \[\Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\gamma\theta_{k_j}}:\big(w-\mu^+_{j}+\tfrac{\delta_\diamond}{2}k_j\big)_+>0\Big\}\Big|=0,\] which also clearly entails \[\begin{align} \label{eq:it1-2c-sup} {\textstyle\mathop{\mathrm{ess\, sup}}_{Q_{\frac{r_j}{4}}^{\gamma\theta_{k_j}}}} w<\mu^+_{j}-\frac{\delta_\diamond}{2}k_j. \end{align}\tag{81}\] We choose \(r_{j+1}:=\lambda r_j\), \(k_{j+1}:=\eta k_j\) and \(\theta_{j+1}:=\theta_{k_{j+1}}\). The properties for \(k_j\) and \(r_j\) are obviously satisfied. Additionally, since \(2\eta>1\) we may follow the same reasoning as before to find that \[\begin{align} \Phi'\big(\Phi^{-1}(k_j)\big)\leq C_1\Phi'\big(\tfrac{1}{2}\Phi^{-1}(k_j)\big)\leq C_1\Phi'\big(\tfrac{1}{2}\Phi^{-1}(2\eta k_j)\big)\leq C_1\Phi'\big(\Phi^{-1}(\eta k_j)\big). \end{align}\] Therefore, in this case \[\begin{align} r_{j+1}^2\theta_{k_{j+1}}=\frac{\gamma r_j^2}{16 C_1\Phi'\big(\Phi^{-1}(\eta k_j)\big)}\leq\frac{\gamma r_j^2}{16}\theta_{k_j}=\Big(\frac{r_j}{4}\Big)^2\gamma\theta_{k_j}\quad\text{and}\quad r_{j+1}\leq \frac{r_j}{4} \end{align}\] and hence \(Q_{j+1}\subseteq Q_{\frac{r_j}{4}}^{\gamma\theta_{k_j}}\subseteq Q_j\). This, together with 81 and \(\mathop{\mathrm{ess\,osc}}_{Q_j}w\leq 4k_j\), yields the estimate \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \textstyle{\mathop{\mathrm{ess\,osc}}_{Q_{\frac{r_j}{4}}^{\gamma\theta_{k_j}}}} w\leq \mu^+_{j}-\frac{\delta_\diamond}{2}k_j-\mu^-_j\leq 4k_j-\frac{\delta_\diamond}{2}k_j=4\big(1-\frac{\delta_\diamond}{8}\big)k_j=4k_{j+1}, \end{align}\] finishing the proof in this case and also the proof of the lemma as a whole. ◻

Proof:. The proof is similar to the previous lemma. This time, using the fact that \((k_j)_{j\in\mathbb{N}}\) is non-increasing and that the \((Q_j)_{j\in\mathbb{N}}\) are nested, we always have \(\mu^-_{j}:=\mathop{\mathrm{ess\,inf}}_{Q_j} w> k_j\) inside the cylinder of the current iteration step. Accordingly, whenever we draw on Lemma 8 or Lemma 12 we are in the case \(\theta=\theta_{\mu^-_{j}}\) and we will adjust our choice for the scaling factor of the next cylinder to \(\theta_{j+1}=\theta_{\mu^-_{j}}\) instead of \(\theta_{k_{j+1}}\) as taken in Lemma 13. Notice that by this choice, \(\theta_{j+1}\leq\theta_j\) and \(Q_{j+1}\subseteq Q_j\) are evident in most of the cases, as \((r_j)_{j\in\mathbb{N}}\) and \((\theta_j)_{j\in\mathbb{N}}\) are non-increasing, the latter due to \(\mu^-_{j-1}\leq\mu^-_{j}\), so that \(r_{j+1}^2\theta_{j+1}=r_{j+1}^2\theta_{\mu^-_{j}}\leq r_j^2\theta_{\mu^-_{j-1}}=r_j^2\theta_j\). Aside from the switch in scaling factors, the general steps are quite similar to the previous lemma, nevertheless, let us briefly verify the main steps.

Again we let the numbers \(\varpi_0=\varpi_0(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac12)\) and \(\delta_\diamond=\delta_\diamond(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac14)\) be given by Lemma 8 and Lemma 12, respectively, and claim that the lemma holds if we take the same \(\lambda=\lambda(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac12)\), \(\eta=\eta(\Phi,M,q_1,q_2,a,\hat{b})\in(\tfrac{31}{32},1)\), \(\beta<1\) and \(C=C(\Phi,M,q_1,q_2,a,\hat{b})\geq 1\) as in 79 . Now, assume 82 holds for some \(j\in\mathbb{N}\). For the choice for \(r_{j+1}\), \(k_{j+1}\) and \(\theta_{j+1}\) we discern the following cases:

Case 1: \(\mathop{\mathrm{ess\,osc}}_{Q_j} w<2 k_j\).
Pick \(r_{j+1}:=\frac{r_j}{2}\), \(k_{j+1}:=\frac{k_j}{2}\), \(\theta_{j+1}:=\theta_{\mu^-_{j}}\). As discussed above \(\theta_{j+1}\leq \theta_{j}\) and \(Q_{j+1}\subseteq Q_j\) follows from \(r_{j+1}\leq r_j\). Thus, \(\mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq\mathop{\mathrm{ess\,osc}}_{Q_j} w<2k_j=4k_{j+1}\). The inequalities for \(r_{j+1}\) and \(k_{j+1}\) are obvious.

Case 2: \(2k_j\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4k_j\).
Again we split this case in accordance with the Lemmas 8 and 12. The lemmas will be employed for \(\vartheta=\theta_j\), \(r=\frac{r_j}{2}\), \(k=k_j\), which is possible due to \(k_j<\mu^-_{j}\) and \(\theta_{j+1}\leq\theta_{j}\) implying that here \(\theta_j= \vartheta\geq \theta=\theta_{\mu^-_{j}}=\theta_{j+1}\).

Case 2a: \(\delta_\diamond^2k_j^2\theta_{j+1}^{-\frac{q_1-1}{q_1}}\leq \big(\tfrac{r_j}{2}\big)^{N\kappa}\).
Set \(r_{j+1}:=\frac{r_j}{2}\), \(k_{j+1}:=k_j\), \(\theta_{j+1}:=\theta_{\mu^-_j}\). The arguments for \(\theta_{j+1}\leq\theta_j\), \(Q_{j+1}\subseteq Q_j\) and \(\mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\) remain the same as in the previous case. That the properties for \(r_{j+1}\) and the lower bound for \(k_j\) are satisfied is also obvious. Let us check the upper bound for \(k_{j+1}\). Since \(k_j\leq k_{j-1}\leq\dots \leq k_0<\mu^-_{0}\leq \dots\leq\mu^-_{j}\) we have \(\theta_{k_j}\geq\theta_{k_0}=\theta_{0}\geq \theta_{j+1}\) and therefore the case assumption also implies that \(\delta_\diamond^2 k_j^2\theta_{k_j}^{-\frac{q_1-1}{q_1}}\leq\big(\frac{r_j}{2} \big)^{N\kappa}\) and we can copy the steps presented in Case 2a of Lemma 13.

Case 2b: \(\delta_\diamond^2k_j^2\theta_{j+1}^{-\frac{q_1-1}{q_1}}> \big(\tfrac{r_j}{2}\big)^{N\kappa}\) and \(\big|\big\{(x,t)\in Q_{\frac{r_j}{2}}^{\theta_{j+1}}:(w-\mu^-_{j}-k_j)_->0\big\}\big|\leq\varpi_0\big|Q_{\frac{r_j}{2}}^{\theta_{j+1}}\big|\).
We choose \(r_{j+1}:=\frac{r_j}{4}\), \(k_{j+1}:=\frac{7}{8}k_j\) and \(\theta_{j+1}:=\theta_{\mu^-_{j}}\). From \(\delta_\diamond<\frac{1}{4}\) we find that the assumption in this case implies \(k_j\theta_{j+1}^{-\frac{q_1-1}{q_1}}>\big(\frac{r_j}{2}\big)^{N\kappa}\) and in view of Lemma 8 we therefore obtain \[\begin{align} \Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\theta_{j+1}}:\big(w-\mu^-_{j}-\tfrac{k_j}{2}\big)_->0\Big\}\Big|=0\quad\text{and thus }\quad{\textstyle\mathop{\mathrm{ess\,inf}}_{Q_{\frac{r_j}{4}}^{\theta_{j+1}}}} w>\mu^-_{j}+\frac{k_j}{2}, \end{align}\] so that we may conclude analogously to Case 2b of Lemma 13 if we show \(Q_{j+1}\subseteq Q_{\frac{r_j}{4}}^{\theta_{j+1}}\subseteq Q_{j}\). The first inclusion is clear from the choice of \(r_{j+1}\) since the cylinders have the same scaling factor, whereas the second inclusion relies on \(\theta_{j+1}\leq\theta_{j}\).

Case 2c: \(\delta_\diamond^2k_j^2\theta_{j+1}^{-\frac{q_1-1}{q_1}}> \big(\tfrac{r_j}{2}\big)^{N\kappa}\) and \(\big|\big\{(x,t)\in Q_{\frac{r_j}{2}}^{\theta_{j+1}}:(w-\mu^+_{j}+k_j)_+>0\big\}\big|\leq(1-\varpi_0)\big|Q_{\frac{r_j}{2}}^{\theta_{j+1}}\big|\).
Pick \(r_{j+1}:=\lambda r_j\), \(k_{j+1}:=\eta k_j\), \(\theta_{j+1}:=\theta_{\mu^-_{j}}\) and let \(\gamma:=\frac{\varpi_0}{2}\in(0,\tfrac14)\). In view of Lemma 12 we have \[\begin{align} \Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\gamma\theta_{j+1}}:\big(w-\mu^+_{j}+\tfrac{\delta_\diamond}{2}k_j\big)_+>0\Big\}\Big|=0\quad\text{and thus}\quad {\textstyle\mathop{\mathrm{ess\, sup}}_{Q_{\frac{r_j}{4}}^{\gamma\theta_{j+1}}}} w<\mu^+_{j}-\frac{\delta_\diamond}{2}k_j. \end{align}\] Since \(\lambda\leq \frac{\sqrt{\gamma}}{4}<\frac{1}{8}\) we have \[\begin{align} r_{j+1}^2\theta_{j+1}\leq\frac{\gamma r_j^2}{16}\theta_{j+1}=\Big(\frac{r_j}{4}\Big)^2\gamma\theta_{j+1}<r_j^2\theta_j, \end{align}\] which together with \(r_{j+1}\leq \frac{r_j}{4}\leq r_j\) implies \(Q_{j+1}\subseteq Q_{\frac{r_j}{4}}^{\gamma\theta_{j+1}}\subseteq Q_j\). Thus, \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}}w\leq \textstyle{\mathop{\mathrm{ess\,osc}}_{Q_{\frac{r_j}{4}}^{\gamma\theta_{j+1}}}}w\leq \mu^+_{j}-\tfrac{\delta_\diamond}{2}k_j-\mu_j^-\leq 4k_j-\tfrac{\delta_\diamond}{2}k_j=4\eta k_j=4k_{j+1}, \end{align}\] as intended, finishing the proof. ◻

Proof:. Let \(Q_0:=Q_{r_0}^{\theta_{k_0}}\). Then \(\mathop{\mathrm{ess\,osc}}_{Q_0} w\leq 2M=4k_0\) and accordingly, \(r_0\), \(Q_0\) and \(k_0\) fulfill the requirements of Lemma 13, providing \(\lambda=\lambda(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac12)\), \(\eta=\eta(\Phi,M,q_1,q_2,a,\hat{b})\in(\tfrac12,1)\), \(\beta=\beta(q_1,q_2)\in(0,1)\), \(C=C(\Phi,M,q_1,q_2,a,\hat{b})\geq 1\) and sequences \((r_j)_{j\in\mathbb{N}}\), \((k_j)_{j\in\mathbb{N}}\), \((\theta_j)_{j\in\mathbb{N}}\) such that with \(Q_j:=Q_{r_j}^{\theta_j}\) either 77 holds for all \(j\in\mathbb{N}\), or there is \(j_\star\in\mathbb{N}\) such that 77 holds for \(j\leq j_\star\) and \(\mu^-_{j_\star}:=\mathop{\mathrm{ess\,inf}}_{Q_{j_\star}}w>k_{j_\star}\). If such a \(j_\star\) exists, then \(r_{j_\star}\), \(Q_{j_\star}=Q_{r_{j_{\star}}}^{\theta_{j_{\star}}}\) and \(k_{j_\star}\) satisfy the conditions of Lemma 14 and we find that 82 also holds for all \(j>j_\star\). Accordingly, we may continue the sequence obtained for \(j\leq j_\star\) and find that, irrespective of the existence of \(j_\star\), \[\begin{align} \lambda r_{j-1}\leq r_j\leq\frac{1}{2}r_{j-1},\quad k_j\leq \max\big\{\eta k_{j-1}, Cr_{j-1}^{\beta}\big\},\quad\mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4k_j \end{align}\] hold for all \(j\in\mathbb{N}\). From these recursive inequalities we conclude that \[\begin{align} \lambda_\star^jr_0\leq r_j\leq 2^{-j}r_0,\qquad\text{and}\qquad\mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4k_j\leq 4\max\big\{k_0\eta^j, C 2^{-(j-1)\beta}r_0^{\beta}\big\}\leq C_\star\eta_\star^j \end{align}\] hold for all \(j\in\mathbb{N}\), where we used \(r_0\leq 1\) and \(k_0\leq M\) and put \(\lambda_\star=\lambda\), \(\beta_\star=\beta\), \(\eta_\star:=\max\{\eta,2^{-\beta}\}\) and \(C_\star:=4\max\{M,2^\beta C\}\). Then, since \(\theta_j\geq \theta_M\) and \(\lambda_\star^jr_0\leq r_j\) for all \(j\in\mathbb{N}\) we have \[Q_{\lambda_\star^jr_0}^{\theta_M}\subseteq Q_{\lambda_\star^j r_0}^{\theta_j}\subseteq Q_{r_j}^{\theta_j}=Q_j\quad\text{for all }j\in\mathbb{N},\] and therefore \[\mathop{\mathrm{ess\,osc}}_{Q_{\lambda_\star^jr_0}^{\theta_M}} w\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w\leq C_\star\eta_\star^j\quad\text{for all }j\in\mathbb{N}.\hfill\qedhere\] ◻

Proof:. Given any compact \(K\subset\Omega_T\) we set \(d_K:=d_p(K,\Gamma)\) and pick \(R:=\frac{d_K}{2(1+\sqrt{\theta_{0}})}\) with \(\theta_0:=\theta_{\frac{M}{2}}\). Given two points \((x_0,t_0),(x_1,t_1)\in K\), where without loss of generality we assume \(t_0>t_1\), we fix the cylinder \(Q_R^{\theta_M}:=Q_R(x_0,t_0-\theta_M R^2,t_0)\). Since \(d_K\leq \min\big\{\frac{1}{\sqrt{N}}\mathop{\mathrm{dist}}(x_0,\mathop{\mathrm{\partial\Omega}}),\sqrt{t_0},1\big\}\) we in particular have \(R\leq \min\big\{\frac{1}{\sqrt{N}}\mathop{\mathrm{dist}}(x_0,\mathop{\mathrm{\partial\Omega}}),\sqrt{\frac{t_0}{4\theta_0}},1\big\}\) and because of \(\theta_{0}>\theta_M\) also \(Q_R^{\theta_M}\subset Q_R^{\theta_0}\subset\Omega_T\). Denote by \(\lambda_\star=\lambda_\star(\Phi,M,q_1,q_2,a,\hat{b})\in(0,\tfrac12)\), \(\eta_\star=\eta_\star(\Phi,M,q_1,q_2,a,\hat{b})\in(\tfrac12,1)\) and \(C_\star=C_\star(\Phi,M,q_1,q_2,a,\hat{b})\geq1\) the numbers from Corollary 5 and set \(\alpha=\alpha(\Phi,M,q_1,q_2,a,\hat{b})=\frac{\log\eta_\star}{\log\lambda_\star}\in(0,1)\). Now consider the following cases:

If \((x_1,t_1)\not\in Q_{\lambda_\star R}^{\theta_M}\) then \(\|x_0-x_1\|_\infty\geq \lambda_\star R\) or \(\theta_M^{-\frac{1}{2}}|t_0-t_1|^\frac{1}{2}\geq \lambda_\star R\). Accordingly, because of \(\|x\|_\infty\leq|x|\), also \(|x_0-x_1|+\theta_M^{-\frac{1}{2}}|t_0-t_1|^\frac{1}{2}\geq \lambda_\star R\) and since \(|w|\leq M\), we can thus estimate \[\begin{align} \big|w(x_0,t_0)-w(x_1,t_1)\big|\leq 2M&\leq 2M\left(\frac{|x_0-x_1|+\theta_M^{-\frac{1}{2}}|t_0-t_1|^\frac{1}{2}}{\lambda_\star R}\right)^\alpha\\ &\leq \frac{2^{1+\alpha}M}{\lambda_\star^\alpha}\big(1+\sqrt{\theta_{0}}\big)^\alpha\big(1+\theta_M^{-\frac{1}{2}}\big)^\alpha\left(\frac{|x_0-x_1|+|t_0-t_1|^\frac{1}{2}}{d_K}\right)^\alpha\\ &\leq \frac{4M}{\lambda_\star}\big(1+\sqrt{\theta_0}\big)\big(1+\theta_M^{-\frac{1}{2}}\big)\left(\frac{|x_0-x_1|+|t_0-t_1|^\frac{1}{2}}{d_K}\right)^\alpha, \end{align}\] where we made use of \(\lambda_\star\leq 1\) and \(\alpha\in(0,1)\).

On the other hand, if \((x_1,t_1)\in Q_{\lambda_\star R}^{\theta_M}\), we can pick \(j\geq 1\) such that \((x_1,t_1)\in Q_{\lambda_\star^j R}^{\theta_M}\) but \((x_1,t_1)\not\in Q_{\lambda_\star^{j+1} R}^{\theta_M}\), meaning that \(\lambda_\star^{j+1}R\leq |x_0-x_1|+\theta_M^{-\frac{1}{2}}|t_0-t_1|^\frac{1}{2}\). Drawing on Corollary 5 and our choice for \(\alpha\) then provides the estimate \[\begin{align} \big|w(x_0,t_0)-w(x_1,t_1)\big|&\leq \mathop{\mathrm{ess\,osc}}_{Q_{\lambda_\star^j R}^{\theta_M}} w\leq C_\star\eta_\star ^j=C_\star\lambda_\star^{j\alpha}=\frac{C_\star \big(\lambda_\star^{j+1}R\big)^\alpha}{\big(\lambda_\star R\big)^{\alpha}}\\ &\leq \frac{C_\star}{\lambda_\star^\alpha}\left(\frac{|x_0-x_1|+\theta_M^{-\frac{1}{2}}|t_0-t_1|^\frac{1}{2}}{R}\right)^\alpha\\ &\leq 2^\alpha\frac{C_\star}{\lambda_\star^\alpha}\big(1+\sqrt{\theta_0}\big)^\alpha\big(1+\theta_M^{-\frac{1}{2}}\big)^\alpha\left(\frac{|x_0-x_1|+|t_0-t_1|^\frac{1}{2}}{d_K}\right)^\alpha\\ &\leq\frac{2C_\star}{\lambda_\star}\big(1+\sqrt{\theta_0}\big)\big(1+\theta_M^{-\frac{1}{2}}\big)\left(\frac{|x_0-x_1|+|t_0-t_1|^\frac{1}{2}}{d_K}\right)^\alpha \end{align}\] and we may conclude the lemma. ◻

Proof of Theorem 1:. The proof of Theorem 1 is an immediate consequence of Lemma 15. ◻

Proof:. We note that the assumed coordinate system coincides with the system described in [43] and that accordingly there are \(\rho_0>0\) and \(\gamma\in C^{1}\!\left(\mathbb{R}^{N-1}\right)\) such that \[B_{\rho_0}(x_0)\cap \Omega=\big\{x\in B_{\rho_0}(x_0)\,\vert\, x_N>\gamma(x_1,\dots,x_{N-1})\big\}.\] Recall moreover that the partial derivatives of \(\gamma\) vanish when evaluated at \(\check{x}_0\). Accordingly, by continuity we can find \(\rho_1\in(0,\rho_0)\) with the property that for all \(x\in B_{\rho_1}(x_0)\cap\Omega\) we have \(|\nabla\gamma(\check{x})|\leq \frac{1}{2\sqrt{N}}\). In light of the mean value theorem, we hence find that for all \(0<r\leq r_0:=\min\{\frac{\rho_1}{\sqrt{N}},1\}\) and each \(\check{x}\in\check{\mathcal{K}}_r(x_0)\) we have \(|\gamma(\check{x})|<r\), ensuring that \(\mathop{\mathrm{\partial\Omega}}\) does not leave \(\mathcal{K}_r(x_0)\) through the top or bottom of the cube with respect to the \(x_N\)-direction. Consequently, for \(r\in(0,r_0]\) we obtain 84 . ◻

Proof:. As explained at the start of the section we may without loss of generality assume that 9 is written in the coordinate system fulfilling the conditions of Lemma 16 and let \(r_0\in(0,1]\) and \(\gamma\in C^{1}\!\left(\mathbb{R}^{N-1}\right)\) be provided by Lemma . We fix an arbitrary \(r\in(0,r_0]\) and pick any smooth cutoff function \(\psi\) in \(Q_r\) satisfying \(0\leq\psi\leq 1\) and \(\psi_{\vert\partial_p Q_r}=0\) and set \(\Psi_h=\pm\big(\Phi\big([\Phi^{-1}(w)]_h\big)-K\big)_{\pm}\psi^2\). Noticing that \(\Psi_h\) is an eligible test-function for 12 , we repeat the steps of Lemma 5 and obtain \[\begin{align} 0&=\!\iint\limits_{Q_{r}\cap\Omega_T}\!\!\big(\Lambda^\pm(w)\big)_t\psi^2\pm\!\iint\limits_{Q_{r}\cap\Omega_T}\!\!\big(\nabla w+a(x,t)\Phi^{-1}(w)\big)\cdot\nabla\big((w-K)_\pm\psi^2\big)\mp\!\iint\limits_{Q_{r}\cap\Omega_T}\!\! b\big(x,t,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\\&\mp\int_{T_0}^{T_1}\!\int_{\mathcal{K}_r\cap\mathop{\mathrm{\partial\Omega}}} g\big(x,t,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2=:I_1^\pm+I_2^\pm+I_3^\pm+I_4^\pm \end{align}\] due to \(\psi\) disappearing on \(\partial_p Q_r\) but not on \(\mathop{\mathrm{\partial\Omega}}\). The first three integrals may be treated in exactly the same fashion as in Lemma 5 – while replacing \(A_{K,r}^\pm\) with \(B_{K,r}^\pm\) – using the well-signed term \[\iint\limits_{Q_{r}\cap\Omega_T}\!\big|\nabla\big((w-K)_\pm\psi\big)\big|^2,\] so we only concern ourselves with a detailed treatment of \(I_4^\pm\). In light of 84 we can draw on \(\gamma\) to rewrite the boundary integral in the following way. \[\begin{align} |I_4^\pm|&=\Big|\int_{T_0}^{T_1}\!\!\int_{\mathcal{K}_r\cap\mathop{\mathrm{\partial\Omega}}} g\big(x,\tau,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\mathop{\mathrm{d\!}}S\mathop{\mathrm{d\!}}\tau\Big|\\ &=\Big|\int_{T_0}^{T_1}\!\!\int_{\check{\mathcal{K}}_r} \Big(g\big(x,\tau,\Phi^{-1}(w)\big)\big(w-K)_\pm\psi^2\Big)\Big\vert_{x_N=\gamma(\check{x})}\sqrt{1+|\nabla\gamma(\check{x})|^2}\mathop{\mathrm{d\!}}\check{x}\mathop{\mathrm{d\!}}\tau\Big| \end{align}\] Making use of the assumed extension \(\hat{g}\) of \(g\) onto \(\Omega\) for a.e. \(t\in(0,T)\), the facts that \(\psi=0\) on \(\partial_p Q_r\) and \(|\nabla\gamma|\leq 1\) on \(\check{\mathcal{K}}_r\) and 84 once more, we can further estimate \[\begin{align} |I_4^\pm|&\leq\sqrt{2}\int_{T_0}^{T_1}\!\!\int_{\check{\mathcal{K}}_r}\Big|\int_{\gamma(\check{x})}^r\frac{\partial}{\partial x_N}\Big(\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\Big)\mathop{\mathrm{d\!}}x_N\Big|\mathop{\mathrm{d\!}}\check{x}\mathop{\mathrm{d\!}}\tau\\ &\leq \sqrt{2}\int_{T_0}^{T_1}\!\!\int_{\mathcal{K}_r\cap\,\Omega}\Big|\frac{\partial}{\partial{x_N}}\Big(\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)(w-K)_\pm\psi^2\Big)\Big|\mathop{\mathrm{d\!}}x\mathop{\mathrm{d\!}}\tau. \end{align}\] Hence, drawing on 10 and Young’s inequality, we have \[\begin{align} |I_4^\pm|&\leq \sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}\big|\partial_{x_N}\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)\big|(w-K)_\pm\psi^2+\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}\big|\partial_{u}\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)\big|\big|\nabla\Phi^{-1}(w)\big|(w-K)_\pm\psi^2\\&\quad+\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}\big|\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)\big|\big|\nabla(w-K)_\pm\big|\psi^2+2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}\big|\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)\big|(w-K)_\pm\psi|\nabla\psi|\\ &\leq \sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau)(w-K)_\pm\psi^2+\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}\frac{g_0(x,\tau)\Phi'\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}|\nabla (w-K)_\pm|(w-K)_\pm\psi^2\\ &\qquad \sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}g_0(x,\tau)w|\nabla(w-K)_\pm|\psi^2+2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}g_0(x,\tau) w(w-K)_\pm\psi|\nabla\psi|\\ &\leq \sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau)(w-K)_\pm\psi^2+\frac{1}{4}\iint\limits_{Q_{r}\cap\Omega_T}\big|\nabla(w-K)_\pm\big|^2\psi^2+4\iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau)(w-K)_\pm^2\psi^2\\ &\qquad+8\iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau) w^2\psi^2\chi_{[(w-K)_\pm>0]}+\frac{1}{2}\iint\limits_{Q_{r}\cap\Omega_T}(w-K)_\pm^2|\nabla\psi|^2, \end{align}\] where \(\chi_{[(w-K)_\pm>0]}\) denotes the characteristic function of the set \(\{(w-K)_\pm>0\}\). Since \((w-K)_\pm\leq 2M\) and \(w\leq M\), we can treat the integrals containing \(g_0\) with the same methods we used for the integrals containing \(a\) and \(b\) in Lemma 5. E.g., using two applications of Hölder ’s inequality in the third integral on the right leads to \[\begin{align} \iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau)(w-K)_\pm^2\psi^2&\leq 4M^2\!\!\iint\limits_{Q_{r}\cap\Omega_T}g_0^2(x,\tau)\chi_{[(w-K)_\pm>0]}\leq\int_{T_0}^{T_1}\!\Big(\int_{B_{K,r}^\pm(\tau)} g_0^{2q_2}(x,\tau)\Big)^\frac{1}{q_2}\big|B_{K,r}^\pm(\tau)\big|^\frac{q_2-1}{q_2}\\&\leq \|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2\Big(\int_{T_0}^{T_1}\big|B_{K,r}^\pm(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}. \end{align}\] Treating the remaining integrals in a similar fashion and gathering everything finally leads to 85 . ◻

Proof:. As in the previous lemma, we again assume without loss of generality that 9 is written in a coordinate system satisfying the conditions of Lemma 16 and that \(r_0\in(0,1]\) and \(\gamma\in C^{1}\!\left(\mathbb{R}^{N-1}\right)\) are given accordingly. Following the testing procedures from Lemma 6 and slightly adjusting the coefficients used in the applications of Young’s inequality to keep a bit more of the well-signed terms stemming from the diffusion, we again only have to consider how to treat the newly arising boundary integral in \[\begin{align} \label{eq:log-est-bdry-eq1} 0&=\iint\limits_{Q_{r}\cap\Omega_T}\big(\xi^2(w)\zeta^2\big)_t+\iint\limits_{Q_{r}\cap\Omega_T}\big(\nabla w+ a(x,t)\Phi^{-1}(w)\big)\cdot\nabla\Psi(w)-\iint\limits_{Q_{r}\cap\Omega_T}b\big(x,t,\Phi^{-1}(w)\big)\Psi(w)\nonumber\\ &-\int_{T_0}^{T_1}\!\int_{\mathcal{K}_r\cap\mathop{\mathrm{\partial\Omega}}}g\big(x,t,\Phi^{-1}(w)\big)\Psi(w). \end{align}\tag{87}\] Recalling 31 , we find that the well-signed terms from the diffusion are of the form \[\begin{align} J_1:=\iint\limits_{Q_{r}\cap\Omega_T}\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2\quad\text{and}\quad J_2:=\iint\limits_{Q_{r}\cap\Omega_T}\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2. \end{align}\] Repeating the re-expression procedure from the previous Lemma and using the properties of \(\xi\), 10 and the calculations for \(\nabla\Psi(w)\) in 31 , we estimate \[\begin{align} &\left|\int_{T_0}^{T_1}\!\!\int_{\mathcal{K}_r\cap\mathop{\mathrm{\partial\Omega}}}g\big(x,t,\Phi^{-1}(w)\big)\Psi(w)\mathop{\mathrm{d\!}}S\mathop{\mathrm{d\!}}t\right|\\ =\;&\left|\int_{T_0}^{T_1}\!\!\int_{\check{\mathcal{K}}_r} \Big(g\big(x,\tau,\Phi^{-1}(w)\big)\big(w-K)_\pm\Psi(w)\Big)\Big\vert_{x_N=\gamma(\check{x})}\sqrt{1+|\nabla\gamma(\check{x})|^2}\mathop{\mathrm{d\!}}\check{x}\mathop{\mathrm{d\!}}\tau\right|\\ \leq\;&\sqrt{2}\int_{T_0}^{T_1}\!\!\int_{\check{\mathcal{K}}_r}\left|\int_{\gamma(\check{x})}^r\frac{\partial}{\partial_{x_N}}\Big(\hat{g}\big(x,\tau,\Phi^{-1}(w)\big)\Psi(w)\Big)\mathop{\mathrm{d\!}}x_N\right|\mathop{\mathrm{d\!}}\check{x}\mathop{\mathrm{d\!}}\tau\\ \leq\;&\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\partial_{x_N} \hat{g}|\big|\Psi(w)\big|+\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\partial_u \hat{g}|\big|\Psi(w)\big|\frac{|\nabla w|}{\Phi'\big(\Phi^{-1}(w)\big)}+\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\hat{g}|\big|\nabla\Psi(w)\big|\\ =\;&2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\partial_{x_N} \hat{g}|\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta^2+2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\partial_u \hat{g}| \xi(w)\xi'(w)\zeta^2|\nabla w|\\&+2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\hat{g}|\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|+2\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\hat{g}|\Phi'\big(\Phi^{-1}(w)\big)\big(1+\xi(w)\big)\big(\xi'(w)\big)^2\zeta^2|\nabla w|\\&+4\sqrt{2}\iint\limits_{Q_{r}\cap\Omega_T}|\hat{g}|\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\xi'(w)\zeta|\nabla\zeta|=I_1+\dots+I_5. \end{align}\] Drawing on the estimates for \(\xi\) and \(\xi'\) in 29 , the monotonicity properties of \(\Phi'\) and \(\Phi^{-1}\) and recalling that \(\xi(w)=0\) on \(\{w\leq L-K\}\) we obtain from Hölder’s inequality that \[\begin{align} I_1&\leq 2\sqrt{2}\Phi'\big(\Phi^{-1}(M)\big)\frac{\ln\big(\frac{1}{\delta}\big)}{\delta K}\iint\limits_{Q_{r}\cap\Omega_T}g_0^2\,\chi_{[w>L-K]}\\&\leq 2\sqrt{2}\Phi'\big(\Phi^{-1}(M)\big)\frac{\ln\big(\frac{1}{\delta}\big)}{\delta K}\|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2\Big(\int_{T_0}^{T_1}\big|B^+_{L-K,r}(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}. \end{align}\] Relying on Young’s inequality, we also have \[\begin{align} I_2&\leq \tilde{\eta}_1\iint\limits_{Q_{r}\cap\Omega_T}\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\big(\xi'(w)\big)^2\zeta^2|\nabla w|^2+\frac{2}{\tilde{\eta}_1}\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\zeta^2\\&\leq \tilde{\eta}_1 J_2+\frac{2\Phi'\big(\Phi^{-1}(M)\big)}{\tilde{\eta}_1}\frac{M\ln\big(\frac{1}{\delta}\big)}{\delta K}\|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2 \Big(\int_{T_0}^{T_1}\!\!\big|B^+_{L-K,r}(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}, \end{align}\] where we also made use of the fact that \(\delta K\leq M\). Similarly, we find that \[\begin{align} I_4&\leq \tilde{\eta}_2 J_2+\frac{2\Phi'\big(\Phi^{-1}(M)\big)}{\tilde{\eta}_2}\frac{1+\ln\big(\frac{1}{\delta}\big)}{\delta^2 K^2}\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2 w^2\chi_{[w>L-K]}\\ &\leq \tilde{\eta}_2 J_2+\!\frac{2\Phi'\big(\Phi^{-1}(M)\big)M^2}{\tilde{\eta}_2}\frac{1+\ln\big(\frac{1}{\delta}\big)}{\delta^2 K^2}\|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2\Big(\int_{T_0}^{T_1}\!\!\big|B^+_{L-K,r}(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1} \end{align}\] and \[\begin{align} I_5&\leq 4\tilde{\eta}_3\iint\limits_{Q_{r}\cap\Omega_T}\Phi'\big(\Phi^{-1}(w)\big)\xi(w)|\nabla\zeta|^2+\frac{2}{\tilde{\eta}_3}\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2w^2\Phi'\big(\Phi^{-1}(w)\big)\xi(w)\big(\xi'(w)\big)^2\zeta^2\\ &\leq 4\tilde{\eta}_3\ln\Big(\frac{1}{\delta}\Big)\iint\limits_{Q_{r}\cap\Omega_T}\Phi'\big(\Phi^{-1}(w)\big)|\nabla\zeta|^2\\&\qquad+\frac{2\Phi'\big(\Phi^{-1}(M)\big)M^2}{\tilde{\eta}_3}\frac{\ln\big(\frac{1}{\delta}\big)}{\delta^2 K^2}\|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2\Big(\int_{T_0}^{T_1}\!\!\big|B^+_{L-K,r}(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}. \end{align}\] For \(I_3\) first note that due to the mean-value theorem, \(\Phi^{-1}(0)=0\) and the monotonicity properties of \(\Phi'\) and \(\Phi^{-1}\) we have \[\frac{1}{\Phi'(\Phi^{-1}(2M))}\leq\frac{\Phi^{-1}(s)}{s}\quad \text{for all }s\in(0,2M),\] so that with \(C_1=C_1(\Phi,M):= M\Phi'\big(\Phi^{-1}(2M)\big)\) we certainly have \(s^2\leq C_1\Phi^{-1}(s)\) for \(s\in[0,M]\). Moreover, recall that by Lemma 1 there is \(C_2=C_2(\Phi,M)>0\) such that \[\Phi^{-1}(s)\frac{\Phi''\big(\Phi^{-1}(s)\big)}{\Phi'\big(\Phi^{-1}(s)\big)}\leq C_2\quad\text{for all }s\in[0,M].\] Hence, we can treat \(I_3\) with Young’s inequality in the following way: \[\begin{align} I_3&\leq \tilde{\eta}_4\iint\limits_{Q_{r}\cap\Omega_T}\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2+\frac{2}{\tilde{\eta}_4}\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2 w^2\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2\\ &\leq \tilde{\eta}_4\iint\limits_{Q_{r}\cap\Omega_T}\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2+\frac{2C_1}{\tilde{\eta}_4}\!\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2 \Phi^{-1}(w)\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2\\ &\leq \tilde{\eta}_4\iint\limits_{Q_{r}\cap\Omega_T}\frac{\Phi''\big(\Phi^{-1}(w)\big)}{\Phi'\big(\Phi^{-1}(w)\big)}\xi(w)\xi'(w)\zeta^2|\nabla w|^2+\frac{2C_1C_2}{\tilde{\eta}_4}\frac{\ln\big(\frac{1}{\delta}\big)}{\delta K}\!\iint\limits_{Q_{r}\cap\Omega_T}|g_0|^2 \chi_{[w>L-K]}\\ &\leq \tilde{\eta}_4 J_1+\frac{2C_1C_2}{\tilde{\eta}_4}\frac{\ln\big(\frac{1}{\delta}\big)}{\delta K}\|g_0\|_{L^{2q_1\;\!}\!\left((0,T);L^{2q_2}(\Omega)\right)}^2\Big(\int_{T_0}^{T_1}\!\big|B^+_{L-K,r}(\tau)\big|^\frac{q_1(q_2-1)}{q_2(q_1-1)}\Big)^\frac{q_1-1}{q_1}. \end{align}\] Gathering the estimates above and treating the non-boundary integrals in 87 as in Lemma 6 – this time choosing \(\eta_1=\frac{1}{4}\), \(\eta_2=\frac{1}{2}\), \(\eta_3=\frac{1}{4}\), \(\eta_4=\frac{1}{4}\) – we can pick \(\tilde{\eta}_1=\frac{1}{4}\), \(\tilde{\eta}_2=\frac{1}{4}\), \(\tilde{\eta}_3=\frac{1}{2}\), \(\tilde{\eta}_4=1\) and finally conclude 86 . ◻

Proof:. From \(\gamma(x_0)=0\) and \(|\nabla\gamma|\leq\frac{1}{2\sqrt{N}}\) in \(\check{\mathcal{K}}_{r}(x_0)\) we find in view of the mean value theorem that \[\begin{align} \label{eq:deformed-DeGiorgi-eq1} |\gamma(\check{x})|=|\gamma(\check{x})-\gamma(\check{x}_0)|\leq\max_{\check{\xi}\in\check{\mathcal{K}}_{r}(x_0)}\big|\nabla\gamma(\check{\xi})\big||\check{x}-\check{x}_0|\leq \frac{r}{2} \quad\text{for all }\check{x}\in\check{\mathcal{K}}_{r}(x_0). \end{align}\tag{88}\] We note that with \(F:\mathcal{K}_{r}(x_0,\gamma)\to\mathbb{R}^N\) given by \(F(x)=\big(x_1,\dots,x_{N-1},r\frac{x_N-\gamma(\check{x})}{r-\gamma(\check{x})}\big)^T\) the image of the deformed cube under \(F\) is a flat half-cube, i.e. \(F\big(\mathcal{K}_r(x_0,\gamma)\big)=\mathcal{K}_{r}(x_0,0)\). Moreover, \(F\) is invertible on \(\mathcal{K}_r(x_0,\gamma)\) and the inverse \(F^{-1}:\mathcal{K}_r(x_0,0)\to\mathcal{K}_r(x_0,\gamma)\) is explicitly given by \(F^{-1}(y)=\big(y_1,\dots,y_{N-1},\frac{r-\gamma(\check{y})}{r}y_N+\gamma(\check{y})\big)^T\). Both \(F\) and \(F^{-1}\) are continuously differentiable and, recalling 88 , the Jacobian determinants satisfy \[\frac{1}{2}\leq \big|\det DF(x)\big|=\frac{r}{r-\gamma(\check{x})}\leq 2\quad\text{for all }x\in\mathcal{K}_r(x_0,\gamma)\] and \[\frac{1}{2}\leq \big|\det DF^{-1}(y)\big|=\frac{r-\gamma(\check{y})}{r}\leq 2\quad\text{for all }y\in\mathcal{K}_r(y_0,0),\] respectively. With the coordinate transform given by \(y=F(x)\) and the Jacobian determinant bounds we now find that \[\begin{align} \label{eq:deformed-DeGiorgi-eq2} \big|\big\{x\in\mathcal{K}_r(x_0,\gamma):\varphi(x)>L\big\}\big| &=\int_{\{y\in\mathcal{K}_r(x_0,0)\,:\,\varphi(F^{-1}(y))>L\}}\big|\det DF^{-1}(y)\big|\mathop{\mathrm{d\!}}y\nonumber\\ &\leq 2\Big|\big\{y\in\mathcal{K}_r(x_0,0):\varphi\big(F^{-1}(y)\big)>L\big\}\Big| \end{align}\tag{89}\] and, along similar lines, \[\begin{align} \label{eq:deformed-DeGiorgi-eq3} \Big|\big\{y\in\mathcal{K}_r(x_0,0)\,:\,\varphi\big(F^{-1}(y)\big)<K\big\}\Big|\geq\frac{1}{2}\big|\big\{x\in\mathcal{K}_r(x_0,\gamma)\,:\,\varphi(x)<K\big\}\big|. \end{align}\tag{90}\] Since \(\mathcal{K}_r(x_0,0)\) is convex, we may draw on Lemma 2 (together with [33]) to find \(C_1=C_1(N)>0\) such that \[\begin{align} &(L-K)\Big|\big\{y\in\mathcal{K}_r(x_0,0):\varphi\big(F^{-1}(y)\big)>L\big\}\Big|\\\leq\;&\frac{C_1r^{N+1}}{\big|\big\{y\in\mathcal{K}_r(x_0,0):\varphi(F^{-1}(y))<K\big\}\big|}\int_{\{y\in\mathcal{K}_r(x_0,0)\,:\,L<\varphi(F^{-1}(y))<K\}}\big|\nabla\varphi\big(F^{-1}(y)\big)\big|\mathop{\mathrm{d\!}}y, \end{align}\] wherein substituting back and using the upper bound on \(|\det DF(x)|\) entails \[\begin{align} \label{eq:deformed-DeGiorgi-eq4} &(L-K)\Big|\big\{y\in\mathcal{K}_r(x_0,0):\varphi\big(F^{-1}(y)\big)>L\big\}\Big|\nonumber\\\leq\;&\frac{2C_1r^{N+1}}{\big|\big\{y\in\mathcal{K}_r(x_0,0)\,:\,\varphi(F^{-1}(y))<K\big\}\big|}\int_{\{x\in\mathcal{K}_r(x_0,\gamma)\,:\,L<\varphi(x)<K\}}\big|\nabla\varphi(x)\big|\mathop{\mathrm{d\!}}x. \end{align}\tag{91}\] Combining 91 with the estimates in 89 and 90 we conclude upon choosing \(C=8C_1\). ◻

Proof:. Keeping close to the reasoning of Lemmas 5 and 17 we test 9 against \(\pm(w-K)_\pm\zeta^2\) and only note that in view of the level restrictions in 92 we find that \(K\geq \mathop{\mathrm{ess\, sup}}_{\mathcal{K}_r}w_0\) entails \((w_0-K)_+=0\) in \(\mathcal{K}_r\cap\,\Omega\) and that \(K\leq \mathop{\mathrm{ess\,inf}}_{\mathcal{K}_r}w_0\) implies \((w_0-K)_-=0\) in \(\mathcal{K}_r\cap\,\Omega\), so that in both of the cases we have \(\Lambda^\pm(w_0)=0\) in \(\mathcal{K}_r\cap\,\Omega\). The remaining terms can be estimated as before. ◻

Proof:. Note that because of \(L-K\geq \mathop{\mathrm{ess\, sup}}_{\mathcal{K}_{r}(x_0)}w_0\) we have \(\big(w_0(x)-L+K\big)_+= 0\) for all \(x\in\mathcal{K}_r\cap\,\Omega\) and accordingly \(\xi(w_0)= 0\) throughout \(\mathcal{K}_r\cap\,\Omega\). Hence, closely following the proofs of Lemma 6 and Lemma 18 we may prove 93 . ◻

Proof:. With \(r_0\), \(k_0\) and \(\theta_{k_0}=\big(\Phi'(\Phi^{-1}(k_0))\big)^{-1}\) provided we set \(Q_0:=Q_{r_0}^{\varpi_0,\theta_{k_0}}\cap\overline{\Omega_T}\), \(\mu^-_{0}:=\mathop{\mathrm{ess\,inf}}_{Q_0} w\) and \(\mu^+_{0}:=\mathop{\mathrm{ess\, sup}}_{Q_0}w\). If \(\mu^-_{0}>k_0\) we set \(j_\star=j_0\) and have nothing to proof. Assume now \(\mu^-_{0}\leq k_0\). For \(j>0\) we denote by \(\mu^-_{j}:=\mathop{\mathrm{ess\,inf}}_{Q_j} w\) and \(\mu^+_{j}:=\mathop{\mathrm{ess\, sup}}_{Q_j}w\) the infimum and supremum of \(w\) in the current cylinder \(Q_j\) and by \(\mu^-_{j,0}:=\mathop{\mathrm{ess\,inf}}_{\mathcal{K}_{j}} w_0\) and \(\mu^+_{j,0}:=\mathop{\mathrm{ess\, sup}}_{\mathcal{K}_{j}}w_0\) the infimum and supremum of the initial datum in the corresponding spatial box centered at \(x_0\). If at some point \(\mu^-_{j}>k_j\) the argument is finished. If not, we will generate \(r_{j+1}\), \(k_{j+1}\), \(\theta_{j+1}\) depending on which of the different cases we find ourselves in. Let \(C_1=C_1(\Phi,2,M)>1\) again be taken from Lemma 1 such that \[\begin{align} \Phi'\big(\Phi^{-1}(s)\big)\leq C_1\Phi'\big(\tfrac12\Phi^{-1}(s)\big)\quad\text{for all }s\in[0,M], \end{align}\] and let \(\delta_\diamond=\delta_\diamond(\Phi,M,q_1,q_2,a,\hat{b},g_0)\in(0,\tfrac14)\) be provided by Lemma 23. We claim that the lemma holds for the choices \[\begin{align} \lambda&:=\frac{1}{4\sqrt{C_1}}\in(0,\tfrac14),\qquad \eta:=\big(1-\tfrac{\delta_\diamond}{32}\big)\in(\tfrac{31}{32},1),\qquad\beta:=\frac{N\kappa}{2+\frac{q_1-1}{q_1}}<1\\ \text{and} \quad C&:=\max\Big\{1,\Big(\big(\Phi^{-1}(M)\big)^{\frac{q_1-1}{q_1}}\delta_\diamond^{-2}\Big)^\frac{1}{2+\frac{q_1-1}{q_1}}\Big\}\geq 1. \end{align}\] We will now construct the next cylinder in accordance with the following cases. Since the properties of \(r_{j+1}\), \(k_{j+1}\) and the inclusion of \(Q_{j+1}\) in \(Q_j\) can be easily adapted from Lemma 13, we will mostly only present details for the inequality with the essential oscillation in 96 .

Case 1: \(\mathop{\mathrm{ess\,osc}}_{Q_j} w<2k_j\).
Set \(r_{j+1}:=\frac{r_j}{2\sqrt{C_1}}\), \(k_{j+1}:=\frac{k_j}{2}\), \(\theta_{j+1}=\theta_{k_{j+1}}\). Then, due to the choice of \(C_1\), we have \(Q_{j+1}\subset Q_j\) and thus \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w< 2k_j=4k_{j+1}\leq \max\Big\{4k_{j+1},8\mathop{\mathrm{ess\,osc}}_{K_{j}}w_0\Big\}. \end{align}\]

Case 2: \(2k_j\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w\leq 4k_j\).
Here, we further separate the case according to the size difference between \(k_j\) and the oscillation of \(w_0\) in \(\mathcal{K}_j\).

Case 2.1: \(k_j< 2\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\).
Set \(r_{j+1}:=\frac{r_j}{2\sqrt{C_1}}\), \(k_{j+1}:=\frac{k_j}{2}\), \(\theta_{j+1}=\theta_{k_{j+1}}\). Again \(Q_{j+1}\subset Q_j\) and hence \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \mathop{\mathrm{ess\,osc}}_{Q_j} w \leq 4 k_j<8 \mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\leq\max\Big\{4k_{j+1},8\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\Big\}. \end{align}\]

Case 2.2: \(2\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\leq k_j\).
In this subcase, we also split the case with regards to the level restrictions of Lemma 20.

Case 2.2a: \(\mu_j^+-\frac{k_j}{4}\leq \mu_{j,0}^+\) and \(\mu_j^-+\frac{k_j}{4}\geq\mu_{j,0}^-\).
This case cannot occur. Notice that substracting the second inequality of Case 2.2a from the first would entail that \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_j} w\leq\mathop{\mathrm{ess\,osc}}_{K_j}w_0+\frac{k_j}{2}\stackrel{\text{Case 2.2}}{\leq} k_j, \end{align}\] which contradicts the assumption of Case 2.

Case 2.2b: \(\mu_j^-+\frac{k_j}{4}<\mu_{j,0}^-\).
In this case, \(K:=\mu_j^-+\frac{k_j}{4}<\mu_{j,0}^-\) fulfills 92 for \(\Lambda^-\) and Lemma 22 becomes applicable for the choices \(\varpi=\varpi_0\), \(k=\frac{k_j}{4}\) and \(r=\frac{r_j}{2}\). Thus, either \[\begin{align} \label{eq:iteration-1-case-alt} \frac{k_j^2}{16}\theta_j^{-\frac{2q_1-1}{q_2}}\leq\left(\frac{r_j}{2}\right)^{N\kappa}\quad\text{or}\quad \Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\varpi_0,\theta_j}\cap\overline{\Omega_T}\,:\,\big(w-\mu_j^--\tfrac{k_j}{8}\big)_->0\Big\}\Big|=0. \end{align}\tag{97}\] If the first is true we pick \(r_{j+1}=\frac{r_j}{2}\), \(k_{j+1}=k_j\), \(\theta_{j+1}=\theta_{k_{j+1}}\) and follow the reasoning of Lemma 13 Case 2a. In particular, we easily see that then \[\mathop{\mathrm{ess\,osc}}_{Q_{j+1}}w\leq 4k_j=4k_{j+1}\leq\max\Big\{4k_{j+1},8\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\Big\}.\] If the second part in 97 is true, we pick \(r_{j+1}=\lambda r_j\), \(k_{j+1}=\frac{31}{32}k_j\), \(\theta_{j+1}=\theta_{k_{j+1}}\) and obtain \(Q_{j+1}\subset Q_{\frac{r_j}{4}}^{\varpi_0,\theta}\) and hence \[\begin{align} \mathop{\mathrm{ess\,osc}}_{Q_{j+1}}w\leq \mathop{\mathrm{ess\,osc}}_{Q_{\frac{r_j}{4}}^{\varpi_0,\theta_j}} w\leq \mu^+_j-\mu^-_j-\frac{k_j}{8}\leq \frac{31}{8}k_j=4k_{j+1}\leq\max\Big\{4k_{j+1},8\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\Big\}. \end{align}\]

Case 2.2c: \(\mu_j^+-\frac{k_j}{4}>\mu_{j,0}^+\).
Here, Lemma 23 becoms applicable for \(k=\frac{k_j}{4}\) and \(r=\frac{r_j}{2}\). Hence, either \[\begin{align} \frac{\delta_\diamond^2 k_j^2}{16}\theta_j^{-\frac{2q_1-1}{q_2}}\leq\left(\frac{r}{2}\right)^{N\kappa}\quad\text{or}\quad \Big|\Big\{(x,t)\in Q_{\frac{r_j}{4}}^{\varpi_0,\theta_j}\cap\overline{\Omega_T}\,:\,\big(w-\mu_j^++\tfrac{\delta_\diamond}{8}k\big)_+>0\Big\}\Big|=0. \end{align}\] If the first is true, again take \(r_{j+1}=\frac{r_j}{2}\), \(k_{j+1}=k_j\) and \(\theta_{j+1}=\theta_{k_{j+1}}\) and follow Case 2a of Lemma 13. To the contrary, if the second holds, pick \(r_{j+1}=\lambda r_j\), \(k_{j+1}=\eta k_j\) and \(\theta_{j+1}=\theta_{k_{j+1}}\). Then, as in Case 2c of Lemma 13, we have \(Q_{j+1}\subset Q_{\frac{r_j}{4}}^{\varpi_0,\theta_j}\) and \[\mathop{\mathrm{ess\,osc}}_{Q_{j+1}} w\leq \mu_j^+-\frac{\delta_\diamond k_j}{8}-\mu^-_j\leq 4k_j-\frac{\delta_\diamond k_j}{8}=4\eta k_j=4k_{j+1}\leq\max\Big\{4k_{j+1},8\mathop{\mathrm{ess\,osc}}_{\mathcal{K}_j}w_0\Big\}. \qedhere\] ◻

Proof of Theorem 2:. The proof can be completed by combining Lemma 24 with arguments corresponding to Lemma 14 and Corollary 5 and the reasoning outlined in Section 5. ◻