Proof. Indeed, \[\int_Q(u-k)^+\,dxdt= \int_0^\infty \mu(\tilde{M}_\xi)d\xi =\int_k^\infty \mu(\tilde{L}_\xi) d\xi,\;\] where \(\tilde{M}_\xi=\{(t,x):\;(u-k)^+\geq \xi\}=\{(t,x):\;u(t,x)\geq k+\xi\}\). In this case, we infer \(\tilde{M}_\xi=M_{k+\xi}\) and \(\int_Q (u-k)^+\,dxdt= \int_0^{\infty}\mu(M_{k+\xi})\,d\xi\). Make the change of variables \(k+\xi=\tau\). We obtain that \(\int_Q (u-k)^+\,dxdt=\int_k^\infty \mu(M_\tau)\,d\tau\). The case of the set \(L_\xi\) is considered by analogy. ◻

Proof. The inequality follows from the interpolation inequality \(\|u\|_{W_p^s(G)}\leq c \|u\|_{W_p^1(G)}^s \|u\|_{L_p(G)}^{1-s}\) [13] and the embedding \(W_p^s(G)\subset L_r(G)\) (\(r=2n/(n-2s)\)) [13]. It suffices to prove the inequality for \(r=2(n+2)/n\), in this case \(r=2n/(n-2s)\) with \(s=n/(n+2)\) and \[\begin{gather} \|u\|_{L_r(Q)}\leq \Bigl(\int_0^T \|u\|_{L_r(G)}^r\,dt\Bigr)^{1/r}\leq \Bigl(\int_0^T c_1\|u\|_{W_2^s(G)}^r\,dt\Bigr)^{1/r}\leq \\ (c_1c)^{1/r}\Bigl(\int_0^T \|u\|_{W_p^1(G)}^{2} \|u\|_{L_2(G)}^{r(1-s)}\,dt\Bigr)^{1/r}\leq (c_1c)^{1/r} \|u\|_{L_2(0,T;W_2^1(G))}^s \|u\|_{L_\infty(0,T;L_2(G))}^{1-s}. \end{gather}\] ◻

Proof. The idea of the proof is similar to that in Sect. 2 of Ch. 5 in [1] but we use Theorem 1 rather than Theorem 6.1 of Ch. 2 in [1]. Make the change of variables \(u=e^{\lambda t}v\) (\(\lambda >0\)). The function \(v\) is a solution to the equation \[\label{e9} Mu= v_t-\sum_{i,j=1}^n \partial_{x_i}(a_{ij}(t,x,ve^{\lambda t})v_{x_j})+ e^{-\lambda t}b(x,t,ve^{\lambda t},e^{\lambda t}\nabla v)+\lambda v=0.\tag{17}\] The initial and boundary conditions are written as \[\label{e10} v|_{t=0}=u_0,\;\;v|_{S}= g(t,x)e^{-\lambda t},\tag{18}\] Let \(w=|v|^m\), \(w^{(k)}=(|v|^m-k)^+\), \(A_k(t)=\{x\in G:\; w(t,x)> k\}\). In this case \(\int_0^T A_k(t)\,dt=\mu(L_k)\), \(L_k=\{(t,x): w(t,x)> k\}\). Multiply 17 by \(|v|^{m-2} v w^{(k)}\) and integrate over \(G\). Choose \(m_0> \max(\|u_0\|_{L_\infty(G)},\|ge^{-\lambda t}\|_{L_\infty(S)}, \tilde{m}_0)\). In view of this choice, \(w^{(k)}|_{\Gamma_T}=0\) for \(k\geq k_0=m_0^m\). Integrating by parts, we obtain that \[\begin{gather} \label{e11} \frac{1}{m}\frac{\partial}{\partial t} \int_G (w^{(k)})^2\,dx +\int_{A_k(t)} \sum_{i,j=1}^n a_{ij}v_{x_j}v_{x_i} [(m-1)|v|^{m-2}w^{(k)}+ m|v|^{2m-2}] \\ + e^{-\lambda t}b(t,x,ve^{\lambda t},e^{\lambda t}\nabla v)|v|^{m-2} v w^{(k)}\,dx=0. \end{gather}\tag{19}\] The condition 12 yields \[\begin{gather} \frac{1}{m}\frac{\partial}{\partial t} \int_G (w^{(k)})^2\,dx + \int_{A_k(t)} \sum_{i,j=1}^m a_{ij}v_{x_j}v_{x_i} [(m-1)|v|^{m-2}w^{(k)}+ m|v|^{2m-2}]+\lambda ww^{(k)}\,dx \leq \\ \int_{A_k(t)} \beta_0a_{ij}v_{x_i}v_{x_j}|v|^{m-2}w^{(k)} + f|v|^{m}w^{(k)} \,dx, \; f=|g_2|+\frac{1}{m_0}g_1. \end{gather}\] This inequality can be rewritten in the form \[\begin{gather} \frac{1}{m}\frac{\partial}{\partial t} \int_G (w^{(k)})^2\,dx + \int_{A_k(t)} \sum_{i,j=1}^m a_{ij}v_{x_j}v_{x_i} [|v|^{m-2}w^{(k)}(m-1-\beta_0) + m|v|^{2m-2}]+\lambda ww^{(k)}\,dx \\ \leq \int_G f|v|^{m}w^{(k)} \,dx,\; \end{gather}\] Next, we take \(m=\beta_0+1\). In this case we arrive at the inequality \[\label{e14} \frac{1}{m}\frac{\partial}{\partial t} \int_{G} (w^{(k)})^2\,dx + \frac{\delta_0}{2m} \int_{G} |\nabla w^{(k)}|^2+ \lambda w w^{(k)}\,dx \leq \int_G f w w^{(k)} \,dx.\tag{20}\] Integrating in \(t\), we conclude that \[\label{e15} \max_{t\in (0,T)} \int_{G} (w^{(k)})^2\,dx + \int_{Q} |\nabla w^{(k)}|^2+ \lambda w w^{(k)}\,dxdt \leq c_0\int_{L_k} f w w^{(k)} \,dxdt,\tag{21}\] where the constant \(c_0\) depends on \(m,\delta_0\). The Hölder inequality implies that \[\int_{L_k} f w w^{(k)} \,dxdt \leq \|f\|_{L_{p}(Q)}\|ww^{(k)}\|_{L_{p'}(L_k)},\;1/p+1/p'=1.\] Since \(ww^{(k)}\leq 2 (w^{(k)})^2+k^2\), we have \(\|ww^{(k)}\|_{L_{p'}(L_k)}\leq c_1( \|w^{(k)}\|_{L_{2p'}(G)}^2+ k^2(\mu(L_k))^{1/p'})\). This inequality, 21 , and the inequality \((w^{(k)})^2\leq ww^{(k)}\) provide the estimate \[\label{e16} \max_{t\in (0,T)} \int_{G} (w^{(k)})^2\,dx + \int_{Q} |\nabla w^{(k)}|^2+ \lambda (w^{(k)})^2\,dx \leq c_2\|w^{(k)}\|_{L_{2p'}(Q)}^2+ c_2k^2 (\mu(L_k))^{1/p'}.\tag{22}\] The Riesz-Thorin theorem or the results in [13] ensure that \(\|w^{(k)}\|_{L_{2p'}(Q)}\leq \|w^{(k)}\|_{L_{r}(Q)}^\theta \|w^{(k)}\|_{L_{2}(Q)}^{1-\theta}\), where \(r=2(n+2)/n\), \(\theta r+(1-\theta)2=2p'\). Since \(p>(n+2)/2\), \(\theta\in (0,1)\). The inequality \[\label{e64} |ab|\leq \varepsilon |a|^p/p+\varepsilon^{1-p} |b|^{p'}/p', \; p'=p/(p-1),\;\varepsilon>0,\tag{23}\] and lemma 1, imply that \[c_2\|w^{(k)}\|_{L_{2p'}(Q)}^2\leq \frac{1}{2}|w^{(k)}|_{1}^2+ c_3\|w^{(k)}\|_{L_{2}(Q)}^2.\] In this case the inequality 22 yields \[\label{e17} \max_{t\in (0,T)} \int_{G} (w^{(k)})^2\,dx + \int_{Q} |\nabla w^{(k)}|^2+ 2\lambda (w^{(k)})^2\,dx \leq 2c_2k^2(\mu(L_k))^{1/p'}+ 2c_3\|w^{(k)}\|_{L_{2}(Q)}^2.\tag{24}\] Note that the constant \(c_3\) is independent of \(\lambda\). Choosing \(\lambda >\max(1,2c_3)\), we conclude that \[\label{e18} \max_{t\in (0,T)} \int_{G} (w^{(k)})^2\,dx + \int_{Q} |\nabla w^{(k)}|^2+ \lambda (w^{(k)})^2\,dx \leq 2c_2k^2 (\mu(L_k))^{1/p'}.\tag{25}\] Next, the Hölder inequality, Lemma 1 and 25 imply that \[\begin{gather} \int_Q w^{(k)}\,dQ \leq \|w^{(k)}\|_{L_{2(n+2)/n}(Q)}(\mu(L_k))^{(n+4)/2(n+2)}\leq c |w^{(k)}|_1 (\mu(L_k))^{(n+4)/2(n+2)}\\ \leq 2c\sqrt{c_2}k(\mu(L_k))^{\alpha}, \; \alpha=(n+4)/2(n+2)+ 1/(2p')>1. \end{gather}\] Thus, we obtain the inequality \[\label{e19} \int_Q w^{(k)}\,dQ \leq c_3 k(\mu(L_k))^{\alpha}, \; \alpha=(n+4)/2(n+2)+ 1/(2p')>1.\tag{26}\] As a consequence, we have \[\label{e195} \int_Q w^{(k_0)}\,dQ \leq c_3 k_0 (\mu(Q))^{\alpha}=M_0, \; \alpha=(n+4)/2(n+2)+ 1/(2p')>1.\tag{27}\] By Theorem 2 \(\int_Q w^{(k)}=\int_k^\infty \mu(L_\tau)\,d\tau=f(k)\). In this case, \(f'(k)=-\mu(L_k)\) and the inequality 25 can be written in the form \(f(k) \leq c_3 k((-f'(k))^{\alpha},\) or in the form \(f'(k)\leq - c_4 f(k)^{1/\alpha}/k^{1/\alpha}\), \(c_4=c_3^{-1/\alpha}\). Integrating this inequality, we arrive at the relation \[(f(k))^{1-1/\alpha}\leq (f(k_0))^{1-1/\alpha} - \frac{c_4 k^{1-1/\alpha}}{1-1/\alpha} + \frac{c_4 k_0^{1-1/\alpha}}{1-1/\alpha}\leq c_5- \frac{c_4 k^{1-1/\alpha}}{1-1/\alpha}\] which ensures that \(f(k)\equiv 0\) for \(k>k_1=(\frac{c_5 (\alpha-1)}{c_4\alpha})^{\alpha/(\alpha-1)}\), \(c_5=(M_0)^{1-1/\alpha}+\frac{c_4 k_0^{1-1/\alpha}}{1-1/\alpha}\). Hence, \(|v|^m\leq k_1\) a.e. ◻

Proof. The proof is in line with that of Theorem 3. Repeating the arguments, we arrive at 19 with an additional summand \(J= -\int_{\Gamma}e^{-\lambda t} \psi(t,x,e^{\lambda t}v) |v|^{m-2}v w^{(k)}\,d\Gamma\) on the right-hand side. Estimate it. In view of the conditions on \(\psi\), we have \[J\leq \int_\Gamma (g_4e^{-\lambda t}+\beta_3|v|)|v|^{m-1}w^{(k)}\,d\Gamma.\] Next, \[\int_G \Delta\rho \beta_3|v| |v|^{m-1}w^{(k)}\,dx + \int_G \nabla \rho \cdot\nabla \beta_3|v|^{m}w^{(k)}\,dx= \int_{\Gamma}\nabla \rho\cdot \nu \beta_3|v|^{m}w^{(k)}\,d\Gamma.\] Therefore, we derive the inequality \[\begin{gather} \int_{\Gamma} \beta_3|v|^{m}w^{(k)}\,d\Gamma \leq 2\int_{\Gamma}\nabla \rho\cdot \nu \beta_3|v|^{m}w^{(k)}\,d\Gamma\leq \\ 2\int_G \Delta\rho \beta_3|v|^{m}w^{(k)}\,dx+ 2\int_G \nabla \rho \cdot\nabla (\beta_3|v|^{m}w^{(k)})\,dx\leq \\ c_1 \int_G \beta_3|v|^{m}w^{(k)}\,dx+ 2\int_G |\nabla (\beta_3|v|^{m}w^{(k)})|\,dx. \end{gather}\] The first summand here is estimated by the quantity \[\label{w21} \int_G \beta_3|v|^{m}w^{(k)}\,dx\leq \beta_3\int_G w w^{(k)}\,dx\leq c_5\int_G (w^{(k)})^2\,dx+ c_5k^2 \mu(A_k(t)).\tag{29}\] The estimate for the second summand is of the form \[\label{w22} \int_G |\nabla \beta_3|v|^{m}w^{(k)}|\,dx\leq \varepsilon \int_{G} |\nabla w^{(k)}|^2 + c(\varepsilon) ((w^{(k)})^2\,dx +k^2 \mu(A_k(t))).\tag{30}\] Next, we use the equality \[\int_G \Delta\rho_1 |v|^{m-2}v w^{(k)}\,dx+ \int_G \nabla \rho_1 \cdot\nabla (|v|^{m-2}v w^{(k)})\,dx= \int_{\Gamma} g_4 |v|^{m-2} v w^{(k)}\,d\Gamma\] which guarantees the relation \[|\int_{\Gamma}e^{-\lambda t} g_4 |v|^{m-2} v w^{(k)}\,d\Gamma| \leq \int_{\Gamma}|\nabla \rho_1| |\nabla ( |v|^{m-2}v w^{(k)})|\,d\Gamma + \frac{1}{m_0}\int_G |\Delta\rho_1| |v|^{m} w^{(k)}\,dx.\] The former summand on the right-hand side of the previous inequality is bounded by \[\label{w24} \int_{G}|\nabla \rho_1| |\nabla |v|^{m-2}v w^{(k)}|\,dx\leq \varepsilon \int_{G} |\nabla w^{(k)}|^2 + c(\varepsilon) (|\nabla \rho_1|^2 (w^{(k)})^2 + k^2\chi_{A_k(t)})\,dx.\tag{31}\] where \(\chi_{A_k(t)}\) is the indicator of the set \(A_k(t)\). Thus, the estimates 31 validate the relation \[\int_0^T \int_Ge^{-\lambda t} g_4 |v|^{m-2} v w^{(k)}\,dxdt \leq \varepsilon \int_{Q}|\nabla w^{(k)}|^2 + c(\varepsilon) \int_Q\tilde{g}_3( (w^{(k)})^2 +k^2\chi_{L_k}(t,x))\,dxdt,\] where \(\tilde{g}_3=|\Delta \rho_1|+|\nabla \rho_1|^2\). By the embedding theorems, \(\tilde{g}_3\in L_p(Q)\). The same arguments as those involved in the estimate of the right-hand side in 21 , ensure the inequality \[\int_Q\tilde{g}_3 (w^{(k)})^2dxdt\leq \varepsilon_1 |w^{(k)}|_1^2 + c(\varepsilon_1) \int_Q |w^{(k)}|^2\,dxdt,\] with \(\varepsilon_1\) an arbitrary positive constant. Moreover, \[k^2 \int_Q \tilde{g}_3 \chi_{L_k(t,x)} \,dxdt\leq k^2\|\tilde{g}_3\|_{L_p(Q)}(\mu(L_k))^{1/p'},\; p'=p/(p-1).\] These inequalities for an appropriate \(\varepsilon_1\) and 29 , 30 validate the estimate \[\int_0^T J\leq 3\varepsilon \int_{Q} |\nabla w^{(k)}|^2 + c(\varepsilon)( \int_Q |w^{(k)}|^2\,dxdt + k^2 (\mu(L_k))^{1/p'}).\] Thus, the inequality 25 from the previous theorem for an appropriate \(\lambda\) and \(\varepsilon\) is replaced with \[\label{w25} \max_{t\in (0,T)} \int_{G} (w^{(k)})^2\,dx + \int_{Q} |\nabla w^{(k)}|^2+ \lambda (w^{(k)})^2\,dx \leq c_4 k^2 (\mu(L_k))^{1/p'}.\tag{32}\] Further, we can repeat the arguments of the previous theorem. ◻

Proof. The claim in the case of the problem 1 , 2 results from Theorem 1.1 of Ch. 5 in [1]. Unfortunately, similar result in the case of the problem 1 , 3 is proven in [1] under more stringent conditions on the data (see Theorem 7.1 of Ch. 5 [1]). So we outline the proof which is in line with that in [1]. Let \(B_r(x_0)\) be a ball of radious \(r\) centered at \(x_0\) and \(Q(\rho,\tau)=\{(t,x): x\in B_{\rho}(x_0), \;t_0<t<t_0+\tau\}\). Fix \((t_0,x_0)\in \overline{Q}\) and multiply the equation \(\eqref{e1}\) by \(u^{(k)}\xi^2(t,x)\), \(u^{(k)}=\max(u-k,0)\), \(k\geq \max_{x\in Q(\rho,\tau)\cap G} u_0(x)\), where \(\xi\) is an arbitrary smooth function with values between zero and 1 and vanishing for \(x\in {\mathbb{R}}^n \setminus B_\rho(x_0)\) for every \(t\). Integrating the equality obtained over \(B_{\rho}(x_0)=B_\rho\) and by parts, we infer \[\begin{gather} \label{e23} \frac{1}{2}\frac{\partial}{\partial t} \int_{B_\rho\cap G} \xi^2 (u^{(k)})^2\,dx +\int_{B_\rho\cap G} \sum_{i,j=1}^n a_{ij}u^{(k)}_{x_j}u^{(k)}_{x_i}\xi^2 + 2\sum_{i,j=1}^n a_{ij}u_{x_j}u^{(k)}\xi \xi_{x_i}+ \\ b(t,x,u,\nabla u) u^{(k)}\xi^2\,dx=-\int_{\Gamma\cap B_\rho}\xi^2 \psi(t,x,u) u^{(k)}\,d\Gamma +\int_{B_\rho} \xi\xi_t(u^{(k)})^2\,dx, \end{gather}\tag{33}\] Let \(A_k(t)=\{x\in B_\rho\cap G: u(t,x)>k\}\), \(L_k=\{(t,x)\in Q(\rho,\tau)\cap Q: u(t,x)>k\}\). The conditions of the theorem yield \[\begin{gather} \frac{1}{2}\frac{\partial}{\partial t} \int_{B_\rho\cap G}\xi^2 (u^{(k)})^2\,dx + \int_{B_\rho\cap G} \delta_0\xi^2 |\nabla u^{(k)}|^2 \leq \int_{B_\rho\cap G} c_1|\nabla u^{(k)}| |\nabla\xi| \xi u^{(k)}\\ +\beta_1|\nabla u^{(k)}|^2 u^{(k)}\xi^2+g_3\xi^2 u^{(k)} \,dx + \int_{\Gamma\cap B_\rho}\xi^2 (g_4+\beta_3|u|)|u^{(k)}|\,d\Gamma + \int_{B_\rho\cap G} |\xi\xi_t|(u^{(k)})^2\,dx. \end{gather}\] Using the inequality \(|\nabla u^{(k)}|\nabla\xi| \xi u^{(k)}\leq \varepsilon_1 |\nabla u^{(k)}|^2\xi^2+ c(\varepsilon_1)|\nabla \xi|^2( u^{(k)})^2\) for sufficiently small \(\varepsilon_1\), we validate the inequality \[\begin{gather} \frac{1}{2}\frac{\partial}{\partial t} \int_{B_\rho\cap G}\xi^2 (u^{(k)})^2\,dx + \frac{2}{3} \int_{B_\rho\cap G} \delta_0\xi^2 |\nabla u^{(k)}|^2\,dx \leq \int_{B_\rho\cap G} \beta_1|\nabla u^{(k)}|^2 u^{(k)}\xi^2+g_3\xi^2 u^{(k)} \,dx \\ \int_{\Gamma\cap B_\rho}\xi^2 (g_4+\beta_3|u|)|u^{(k)}|\,d\Gamma + \int_{B_\rho\cap G} c_1(u^{(k)})^2(|\nabla \xi|^2+\xi|\xi_t|) \,dx. \end{gather}\] Integrating in \(t\) from \(t_0\) to \(t\leq t_0+\tau\) we obtain the inequality \[\begin{gather} \label{s1} \max_{t\in t_0,t_0+\tau} \int_{B_\rho\cap G}\xi^2 (u^{(k)}(t,x))^2\,dx + \frac{2}{3} \int_{L_k} \delta_0\xi^2 |\nabla u^{(k)}|^2\,dxdt \leq \\ \int_{L_k} 2\beta_1|\nabla u^{(k)}|^2 u^{(k)}\xi^2+2g_3\xi^2 |u^{(k)}| \,dxdt + 2\int_{S\cap Q(\rho,\tau)}\xi^2 (g_4+\beta_3|u|)|u^{(k)}|\,dS \\ + 2\int_{L_k} c_1(u^{(k)})^2(|\nabla \xi|^2+\xi|\xi_t|) \,dxdt +\int_{B_\rho\cap G}\xi^2 (u^{(k)}(t_0,x))^2\,dx. \end{gather}\tag{34}\] We have that \((g_4+\beta_3|u|)\leq (g_4+\beta_3M)=\tilde{g}_4\). Arguing as before (see the construction of the function \(\rho_1\)), we can construct a function \(\rho_2\) such that \(\rho_{2t}-\Delta \rho_2=0,\; \frac{\partial \rho_2}{\partial \nu}|_S=\tilde{g}_4\). In this case, we derive the equality \[\int_{Q(\rho,\tau)} \Delta\rho_2 |u_k|\xi^2\,dxdt+ \int_{Q(\rho,\tau)} \nabla \rho_2\nabla |u_k|\xi^2\,dxdt=\int_{S\cap Q(\rho,\tau)} \tilde{g}_4\xi^2 u^{(k)}\,dS\] which ensures the inequality \[\begin{gather} \int_{S\cap Q(\rho,\tau)}\xi^2 (g_4+\beta_3|u|)|u^{(k)}\,dS\leq \int_{L_k}|\Delta \rho_2|\xi^2 |u^{(k)}|+ 2|\nabla\rho_2||\nabla \xi| \xi |u^{(k)}| + \\ |\nabla\rho_2|\xi^2 |\nabla u^{(k)}|\, dxdt \leq \varepsilon \int_{L_k} \xi^2|\nabla u^{(k)}|^2 + c(\varepsilon) \int_{L_k} |u^{(k)}|^2 |\nabla\xi|^2 + (|\Delta \rho_2|+|\nabla \rho_2|^2)\xi^2\,dxdt. \end{gather}\] The Hölder inequality yields \[c(\varepsilon) \int_{L_k} (|\Delta \rho_2|+|\nabla \rho_2|^2)\xi^2\,dxdt\leq c(\int_{L_k}\xi\,dxdt )^{1/p'}.\] So we can write out the estimate \[\begin{gather} \label{s2} \int_{S\cap Q(\rho,\tau)}\xi^2 (g_4+\beta_3|u|)|u^{(k)}|\,dS\leq \varepsilon \int_{L_k} \xi^2|\nabla u^{(k)}|^2 + \\ c(\varepsilon) \int_{L_k} |u^{(k)}|^2 |\nabla\xi|^2\,dxdt + c_1(\varepsilon)(\int_{L_k}\xi\,dxdt )^{1/p'}. \end{gather}\tag{35}\] Similarly, we infer \[\label{s3} \int_{L_k} 2g_3\xi^2 u^{(k)} \,dxdt\leq c_2(M) (\int_{L_k}\xi\,dxdt )^{1/p'}\tag{36}\] Using 35 , 36 , and choosing small \(\varepsilon\), we can rewrite 34 in the form \[\begin{gather} \label{s4} \max_{t\in t_0,t_0+\tau} \int_{B_\rho\cap G}\xi^2 (u^{(k)})^2\,dx + \frac{1}{3} \int_{L_k} \delta_0\xi^2 |\nabla u^{(k)}|^2\,dxdt \leq \int_{L_k} 2\beta_1|\nabla u^{(k)}|^2 u^{(k)}\xi^2\,dxdt \\ + c_3\int_{L_k} c_1(u^{(k)})^2(|\nabla \xi|^2+\xi|\xi_t|) \,dxdt + c_4(\int_{L_k}\xi\,dxdt )^{1/p'}+\int_{B_\rho\cap G}\xi^2 (u^{(k)}(t_0,x))^2\,dx. \end{gather}\tag{37}\] In what follows, we assume that \(k\geq \max_{x\in Q(\rho,\tau)\cap G}u(t,x)-\delta\), where \(\delta=\delta_0/(12\beta_1)\). This assumption proves that \[\begin{gather} \max_{t\in t_0,t_0+\tau} \int_{B_\rho\cap G}\xi^2 (u^{(k)})^2\,dx + \frac{1}{6} \int_{L_k} \delta_0\xi^2 |\nabla u^{(k)}|^2\,dxdt \leq \\ + c_3\int_{L_k} c_1(u^{(k)})^2(|\nabla \xi|^2+\xi|\xi_t|) \,dxdt + c_4(\int_{L_k}\xi\,dxdt )^{1/p'}+\int_{B_\rho\cap G}\xi^2 (u^{(k)}(t_0,x))^2\,dx. \end{gather}\] where \(k\geq \max_{x\in Q(\rho,\tau)\cap G}u(t,x)-\delta\), \(k\geq \max_{x\in Q(\rho,\tau)\cap G} u_0(x)\). Similar inequality is derived for the function \((-u)^{(k)}\). The conditions of Theorem 8.2 of Ch. 2 (see Remark 8.1 after this theorem) in [1] are fulfilled and, thus, the theorem is proven. ◻

Proof. The proof of the estimate ?? is essentially simpler than that of ?? . It relies on the equality \[\varphi(t,x,v_1)-\varphi(t,x,v_2) =\int_0^1\varphi_v(t,x,v_2+\xi(v_1-v_2))\,d\xi(v_1-v_2).\] Hence, we proceed with the proof of ?? . We employ the definition of the norm [13], [14]. The norm in \(W_p^{s_0,2s_0}(S)\) is defined with the use of partition of unity and straightening of the boundary \(\Gamma\). As a result, we can prove the estimate in the simplest case of \(S=(0,\infty)\times \Gamma\), \(\Gamma={\mathbb{R}}^{n-1}\), \(Q=(0,\infty)\times {\mathbb{R}}^n_+\), \({\mathbb{R}}^n_+=\{x\in {\mathbb{R}}^n: \; x_n>0\}\). The norm in this case is defined by the equality (see [13]) \[\begin{gather} \label{ee} \|v(t,x)\|_{W_p^{s_0, 2s_0}(S)}^p=\|v\|_{L_p(S)}^p+ \int_{0}^{\delta} \frac{1}{|\tau|^{1+s_0p}} \|\Delta_{\tau,t} v\|_{L_{p}(S)}^p\,d\tau +\\ \int_{|h|\leq \delta} \frac{1}{|h|^{n-1+2s_0p}} \|\Delta_{h,x} v\|_{L_{p}(S)}^p\,dh=\|v\|_{L_p(S)}^p+ J_1^p(\Delta_{\tau,t} v)+ J_2^p(\Delta_{h,x} v) ,\;\delta>0, \end{gather}\tag{38}\] where \(\Delta_{\tau,t} v=v(t+\tau,x)-v(t,x)\), \(\Delta_{h,x} v=v(t,x+h)-v(t,x)\). Consider the second summand with \(v=a_{ij}(t,x,u)u_{x_j}\) assuming that this function is compactly supported. We have the equality \(\Delta_{\tau,t}v=a_{ij}(t+\tau,x,u(t+\tau,x))\Delta_{\tau,t}u_{x_j}+(a_{ij}(t+\tau,x, u(t+\tau,x))-a_{ij}(t+\tau,x, u(t,x))) u_{x_j}(t,x)+ (a_{ij}(t+\tau,x,u(t,x))-a_{ij}(t,x,u(t,x))) u_{x_j}(t,x)=I_1+I_2+I_3\). The triangle inequality implies that \[J_1(\Delta_{\tau,t} a_{ij} u_{x_j})\leq J_1(I_1)+ J_1(I_2)+J_1(I_3).\] Estimate every summand. We have (see Corollary 1.3 in [18]) \[\begin{gather} \label{in3} J_1(I_1)\leq \|a_{ij}(t+\tau,x,u(t+\tau,x))\|_{L_\infty(S)}\|u_{x_j}\|_{W_p^{s_0}(0,\infty;L_p(\Gamma))}\leq \\ c_0\|a_{ij}(t+\tau,x,u(t+\tau,x))\|_{L_\infty(S)}\|u\|_{W_p^{1,2}(Q)}. \end{gather}\tag{39}\] Next the Hölder inequality yields \[\begin{gather} \label{in4} J_1(I_3)\leq J_1((a_{ij}(t+\tau,x,u(t,x))-a_{ij}(t,x,u(t,x))) u_{x_j}(t,x)\|)\leq \\ \Bigl(\int_0^\delta \frac{1}{|\tau|^{1+s_0p}}\|u_{x_j}\|_{L_{q_5p/(q_5-p)}(S)}^p\|(a_{ij}(t+\tau,x,u)-a_{ij}(t,x,u)\|_{L_{q_5}(S;C([-M,M]))}^p\,d\tau\Bigr)^{1/p}\leq \\ \|a_{ij}(t,x,u)\|_{B_{q_5 p}^{s_0,2s_0} (S;C([-M,M]))} \|u_{x_j}\|_{L_{q_5p/(q_5-p)}(S)}\leq c_1(M)\|u_{x_j}\|_{L_{q_5p/(q_5-p)}(S)}\leq \\ c_2(M) \|u\|_{W_{q_5p/(q_5-p)}^{1/2+\varepsilon/2+1/2\tilde{q}, 1+\varepsilon+1/\tilde{q}}(Q)}^p\leq \|u\|_{W_{p}^{1-\varepsilon_1, 2-2\varepsilon_1}(Q)}^p,\;\tilde{q}=q_5p/(q_5-p), \end{gather}\tag{40}\] where \(\varepsilon_1>0\), \(2\varepsilon_1\leq 1-1/p -(n+1)/q_5-\varepsilon\), \(0<\varepsilon<1-1/p -(n+1)/q_5\) and we employ Corollary 1.3 in [18] and the embedding \(W_{p}^{1-\varepsilon_1, 2-2\varepsilon_1}(Q)\subset W_{q_5p/(q_5-p)}^{1/2+\varepsilon/2+1/2\tilde{q},1+\varepsilon+1/\tilde{q}}(Q)\) (Corollary 5.6.4 of Ch. 7 in [14]). The constant \(\varepsilon\) can be chosen to be arbitrarily small. Estimate the summand \(J_1(I_2)\). We use here the inequality of the Gagliardo-Nirenberg type (Theorem 5.7.1 of Ch. 7 in [14]) \[\label{in5} \|u\|_{B_{q,q}^{\tilde{s}/2,\tilde{s}}(Q)}\leq c\|u\|_{B_{p,p}^{\tilde{s}_1/2,\tilde{s}_1}(Q)}^{\theta} \|u\|_{B_{\infty,\infty}^{\tilde{s}_0/2,\tilde{s}_0}(Q)}^{1-\theta},\tag{41}\] where \(\tilde{s}-(n+2)/q=(1-\theta)\tilde{s}_0+\theta(\tilde{s}_1-(n+2)/p)\), \((\tilde{s}-\tilde{s}_0)/(\tilde{s}_1-\tilde{s}_0)\leq \theta\leq 1\), \(-\infty<\tilde{s}_0<\tilde{s}<\tilde{s}_1<\infty\), \(1\leq q,p\leq \infty\), \(\theta>0\). Since the function \(a_{ij}\) meets the Lipschitz condition in the variable \(u\), we infer \[\begin{gather} \label{in6} J_1(I_2)\leq c(M) J_1(|\Delta_{\tau,t}u(t,x)| |u_{x_j}(t,x)|)\leq c(M)\|u_{x_j}\|_{L_{q}(S)} \|u\|_{B_{qp/(q-p),p}^{{s}_0,2{s}_0}(S)}\leq \\ c_1(M)\|u_{x_j}\|_{W_{q}^{(1+1/q+\varepsilon)/2,1+1/q+\varepsilon}(Q)} \|u\|_{W_{qp/(q-p)}^{{s}_0+(q-p)/2qp+\varepsilon/2,2{s}_0+(q-p)/qp+ \varepsilon}(Q)}, \end{gather}\tag{42}\] where \(\varepsilon>0\) is a small parameter, we use the Hölder inequality, Corollary 1.3 in [18], and the embedding \(B_{q,q}^{s+\varepsilon}\subset B_{q,p}^s\) valid for every \(\varepsilon>0\) and \(p\in [1,\infty]\). To estimate the factors in 42 , we use 41 . Take \(\tilde{s}=1+1/q+\varepsilon\), \(\tilde{s}_0=3\varepsilon\leq \alpha_1\) (see Theorem 5), \(\tilde{s}_1=2-\varepsilon\), \(\theta=1/2\). We can conclude that \[\label{in7} \|u_{x_j}\|_{W_{q}^{1/2+1/2q+\varepsilon/2,1+1/q+\varepsilon}(G)}\leq c \|u\|_{W_p^{1-\varepsilon/2,2-\varepsilon}(Q)}^{1/2} \|u\|_{B_{\infty,\infty}^{\tilde{s}_0/2, \tilde{s}_0}(Q)}^{1/2},\tag{43}\] where \(q=\frac{2p(n+1)}{(n+2)},\) and, thereby, \(1+\frac{1}{q}+\varepsilon-\frac{n+2}{q}= \frac{\tilde{s}_0}{2}+\frac{1}{2} (2-\varepsilon-(n+2)/p\). Now take \(\tilde{s}=1-1/2q=s_0+(q-p)/pq\), \(\tilde{s}_0=2\varepsilon\), \(\tilde{s}_1=2-\varepsilon\). For this choice of \(q\), it follows from 41 that \[\label{in8} \|u\|_{W_{qp/(q-p)}^{1/2-1/2q+\varepsilon/2,1-1/q+\varepsilon}(Q)}\leq c \|u\|_{W_p^{1-\varepsilon/2,2-\varepsilon}(Q)}^{1/2} \|u\|_{B_{\infty,\infty}^{\tilde{s}_0/2, \tilde{s}_0}(Q)}^{1/2},\tag{44}\] where \(1-1/q-(n+2)(1/p-1/q)= \tilde{s}_0/2+ (2-\varepsilon-(n+2)/p)/2.\) In this case the inequality 42 can be written in the form \[\label{in10} J_1(I_2)\leq C(M) \|u\|_{W_p^{1-\varepsilon/2,2-\varepsilon}(Q)}.\tag{45}\] The estimate of the expression \(J_2^p(\Delta_{h,x} v)\) is proven by analogy. ◻

Proof. We present the proof in the case of the problem 1 , 3 . The problem 1 , 2 is simpler and the arguments are the same. Let \(N\in {\mathbb{N}}\) and \(h=T/N\). Construct a finite covering of \(\Gamma\) by balls \(B_h(x_i)\) of radius \(h\), the parameter \(h\) is chosen below. Let us build it up to a covering of \(G\). Next, we can construct the covering of \(Q\) by domains of the form \(Q_{kl}=((k-5/4)h,(k+1/4)h)\times B_h(x_l)\) (\(k=1,2,\ldots, N\)). Construct also the corresponding partition of unity on \(Q\): \(\varphi_{kl}\in C_0^{\infty}(Q_{kl})\). Without loss of generality we assume that \(\sum_{k,l: x_l\in \Gamma}\varphi_{kl}(t,x)=1\) for all \(x\in \Gamma, t\in [0,T]\). Since the function \(u\) satisfies the estimate from Theorem 6 and the functions \(a_{ij}\) are continuous in all variables, for every \(\varepsilon>0\), there exists \(h=T/N>0\) such that \(|a_{ij}(t,x,u(t,x))-a_{ij}(kh,x_l,u(kh,x_l))|<\varepsilon\) for all \((t,x)\in Q_{kl}\) and all \(i,j\), where the constant \(h\) depends on \(M_1\) but it is independent of the function \(u\) itself. Consider the family of problems \[\label{a26} Mv= v_t-\sum_{i,j=1}^n a_{ij}(kh,x_l,u_{kl})v_{x_jx_i}=f,\;\tag{46}\] \[\label{a27} v|_{t=0}=v_0,\;\;v|_{S}= g(t,x),\tag{47}\] or \[\label{a28} v|_{t=0}=u_0,\;\;\sum_{i,j=1}^n a_{ij}(kh,x_l,u_{kl})v_{x_j}\nu_i|_{S}=g(t,x),\tag{48}\] where \(u_{kl}\in [-M,M]\) is a collection of constants and \(M\) is the constant defined in Theorems 3, 4. We suppose that \[\label{a29} v_0\in W_p^{2-2/p}(G),\; g\in W_p^{k_0,2k_0}, \; f\in L_p(Q), \; \Gamma\in C^2,\tag{49}\] where \(k_0=s_1\) in the case of the Dirichlet boundary condition and \(k_0=s_0\) otherwise. Applying Theorem 1 we can say that there exist unique solutions to the problems 46 , 47 and 46 , 48 and the estimates \[\label{a30} \|v\|_{W_{p}^{1,2}(Q)}\leq C_{0}(\|u_{0}\|_{W_{p}^{2-2/p}(G)}+ \|f\|_{L_{p}(Q)}+\|g\|_{W_{p}^{k_{0},2k_{0}}(S)})\tag{50}\] hold, where, without loss of generality, we can assume that \(C_0\) is indepedent of \(k,l\). Multiply the equation 1 and the boundary conditions 3 by \(\varphi_{kl}\). The function \(w=u\varphi_{kl}\) is a solution to the problem \[\begin{gather} \label{e26} Mw= w_t-\sum_{i,j=1}^n a_{ij}(t,x,u)w_{x_jx_i}+ \tilde{b}_{kl}=0,\; \tilde{b}_{kl}= -\varphi_{kl}(\sum_{i,j=1}^n (a_{ijx_i}u_{x_j}+a_{iju}u_{x_j}u_{x_i})- \\ b(x,t,u,\nabla u)) + \sum_{i,j=1}^n a_{ij}u_{x_j}\varphi_{kl x_i}+ \sum_{i,j=1}^n a_{ij}u_{x_i}\varphi_{klx_j}+\sum_{i,j=1}^n a_{ij}u\varphi_{klx_ix_j}, \end{gather}\tag{51}\] \[\label{e27} \sum_{i,j=1}^n a_{ij}(t,x,u)w_{x_j}\nu_i- \sum_{i,j=1}^n a_{ij}(t,x,u)u\varphi _{kl x_j}\nu_i+ \varphi_{kl} \psi(t,x,u)=0, \; \; w(0,x)=\varphi_{kl}u_0.\tag{52}\] Rewrite this problem in the form \[\begin{gather} \label{e28} Mw= w_t-\sum_{i,j=1}^n a_{ij}(kh,x_l,u(k h,x_l))w_{x_jx_i}=-\sum_{i,j=1}^n (a_{ij}(kh,x_l,u(kh,x_l))- \\ a_{ij}(t,x,u(t,x))) w_{x_jx_i})- \tilde{b}_{kl}=\tilde{f}_{kl},\; w(0,x)=\varphi_{kl}u_0. \end{gather}\tag{53}\] \[\begin{gather} \label{e29} \sum_{i,j=1}^n a_{ij}(kh,x_l,u(kh,x_l))w_{x_j}\nu_i=\sum_{i,j=1}^n (a_{ij}(kh,x_l,u(kh,x_l))-a_{ij}(t,x,u(t,x)))w_{x_j}\nu_i+\\ \sum_{i,j=1}^n a_{ij}(t,x,u(t,x)) u\varphi _{klx_j}\nu_i- \varphi_{kl} \psi(t,x,u)=\tilde{g}_{kl},\; w(0,x)=\varphi_{kl}u_0. \end{gather}\tag{54}\] The estimate 50 yields \[\label{e30} \|w\|_{W_{p}^{1,2}(Q)}\leq C_{0}(\|\varphi_{kl}u_0 \|_{W_{p}^{2-2/p}(G)}+ \|\tilde{f}_{kl}\|_{L_{p}(Q)}+\|\tilde{g}\|_{W_{p}^{s_{0},2s_{0}}(S)}).\tag{55}\] Estimate every summand on the right-hand side. We have that \[\label{e31} \|\sum_{i,j=1}^n (a_{ij}(kh,x_l,u(h,x_l))-a_{ij}(t,x,u(t,x)))w_{x_jx_i}\|_{L_p(Q)}\leq c_0(p)\varepsilon \|w\|_{W_p^{1,2}(Q)}.\tag{56}\] Lemma 1.21 in [18] and the conditions on the data imply that \[\begin{gather} \label{e32} \| \varphi_{kl}\sum_{i,j=1}^n a_{ijx_i}u_{x_j}-\sum_{i,j=1}^n a_{ij}u_{x_j}\varphi_{klx_i}- \sum_{i,j=1}^n a_{ij}u_{x_i}\varphi_{klx_j}-\sum_{i,j=1}^n a_{ij}u \varphi_{kl x_ix_j}\|_{L_p(Q)}\leq \\ c\|u\|_{W_p^{1-\sigma,2-2\sigma}(Q)}, \end{gather}\tag{57}\] where \(\sigma\in (0,1)\) is a parameter defined in embedding theorems. Next, we have \[\label{e33} \|\varphi_{kl}(\sum_{i,j=1}^n a_{iju}u_{x_j}u_{x_i}+ b(x,t,u,\nabla u))\|_{L_p(G)}^p\leq c \| \nabla u \|_{L_{2p}(G)}^{2}+ \|g_3\|_{L_p(G)}^p.\tag{58}\] We can assume that \(\alpha_1\leq 2/(n+2)\), otherwise we assign \(\alpha_1=2/(n+2)\). Next, we use the inequality (Lemma 1.14 in [18] or Corollary 5.7.3 of Ch. 7 in [14]) \[\| \nabla u \|_{L_{2p}(G)}\leq c \| \nabla u \|_{C^{2\alpha_1}(\overline{G})}^{1/2} \|u\|_{W_p^{2-2\alpha_1}(G)}^{1/2}\leq c(M_1) \|u\|_{W_p^{2-2\alpha_1}(G)}^{1/2}.\] Integrating 58 in \(t\) and using 57 , we conclude that \[\label{e34} \|\varphi_{kl}(\sum_{i,j=1}^n a_{iju}u_{x_j}u_{x_i}+ b(x,t,u,\nabla u))\|_{L_p(Q)}\leq c_1 \|u \|_{L_p(0,T;W_p^{2-2\alpha_1}(Q))}+ c_2.\tag{59}\] The above estimates imply that \[\label{e35} \|\tilde{f}_{kl}\|_{L_p(Q)}\leq \varepsilon c_0(p) \|w\|_{W_p^{1,2}(Q)} + c(M,M_1) \|u \|_{W_p^{1-\sigma_1,2-2\sigma_1}(Q)},\tag{60}\] where \(\sigma_1=\min(\sigma, \alpha_1)\). Lemma 2 ensures the inequality \[\begin{gather} \label{e36} \|\sum_{i,j=1}^n a_{ij}(t,x,u(t,x)) u\varphi _{ijx_j}\nu_i- \varphi_{ij} \psi(t,x,u)\|_{W_p^{s_0,2s_0}(S)}\leq \\ c_5(M)+ c_6(M)\|u\|_{W_p^{s_1-\varepsilon_0, 2s_1-2\varepsilon_0}(S)}\leq c_5(M)+ c_7 \|u\|_{W_p^{1-\varepsilon_0, 2-2\varepsilon_0}(Q)}, \end{gather}\tag{61}\] with \(\varepsilon_0>0\) -a constant. Moreover, Lemma 2 guarantees the estimate \[\begin{gather} \label{e37} \|\sum_{i,j=1}^n (a_{ij}(kh,x_l,u(kh,x_l))-a_{ij}(t,x,u(t,x)))w_{x_j}\nu_i\|_{W_p^{s_0,2s_0}(S)}\leq \\ c_1\varepsilon \|w\|_{W_p^{1,2}(Q)} + c_7(M) \|w\|_{W_p^{1-\sigma_2,2-2\sigma_2}(Q)}, \end{gather}\tag{62}\] where \(\sigma_2>0\) is a constant. Thus, the estimates 61 and 62 validate the inequality \[\label{e38} \|\tilde{g}_{kl}\|_{W_p^{s_0,2s_0}(S)}\leq c_0 \varepsilon \|w\|_{W_p^{1,2}(Q)} + c_8(M,M_1) \|w\|_{W_p^{1-\sigma_3,2-2\sigma_3}(Q)} + c_9(M,M_1),\tag{63}\] where \(\sigma_3>0\) is a constant. In view of 55 , 60 , 63 , we derive that \[\label{e39} \|w\|_{W_{p}^{1,2}(Q)}\leq (c_0(p)+c_1)C_0 \varepsilon \|w\|_{W_p^{1,2}(Q)} + c_9(M,M_1) \|u\|_{W_p^{1-\sigma_4,2-2\sigma_4}(Q)}+ C_{10}(M,M_1),\tag{64}\] where \(\sigma_4>0\) is a constant. Choose \(\varepsilon<1/(2C_0(c_0(p)+c_1))\) and find the corresponding parameter \(h>0\). We obtain that \[\label{e40} \|w\|_{W_{p}^{1,2}(Q)}\leq 2c_9(M,M_1) \|u\|_{W_p^{1-\sigma_4,2-2\sigma_4}(Q)}+ 2 C_{10}(M,M_1).\tag{65}\] Write out the estimate for a function \(u\) \[\label{e41} \|u\|_{W_{p}^{1,2}(Q)}\leq \sum_{k,l} \|\varphi_{kl}u\|_{W_{p}^{1,2}(Q)} \leq c_{11} \|u\|_{W_p^{1-\sigma_4,2-2\sigma_4}(Q)}+ c_{12}(M,M_1),\;\tag{66}\] where the constant \(C_{12}\) depends on \(M,M_1\) and the corresponding norms of the data, \(\sigma_4\in (0,1)\). Next, we employ the interpolation inequality (Theorem 1.21 in [18] or [14]). \[\|u\|_{W_p^{1-\sigma_4,2-2\sigma_4}(Q)}\leq c_{13} \|u\|_{W_p^{1,2}(Q)}^{\theta}\|u\|_{L_p(Q)}^{1-\theta}\leq \varepsilon \|u\|_{W_p^{1,2}(Q)}+ c_{14}(\varepsilon),\; \theta=1-\sigma_4.\] where \(\varepsilon >0\) is an arbitrary constant. Using this inequality in 66 with \(\varepsilon=1/2C_{11}\), we establish the claim. ◻

Proof. To prove existence theorems, we employ the conventional scheme exposed in Sect. 5,6 of Ch. 5 in [1]. Consider the collections of problems depending on a parameter \(\gamma\in [0,1]\): \[\label{e45} Mu= u_t-\sum_{i,j=1}^n \partial_{x_i}((\gamma a_{ij}(t,x,u)+(1-\gamma)\delta_{ij})u_{x_j})+ \gamma b(x,t,u,\nabla u)=0,\tag{71}\] In the former case the initial-boundary conditions are written in the form \[\label{e46} u|_{t=0}=u_0,\;\;u|_{S}= g(t,x),\tag{72}\] and in the latter in the form \[\label{e47} u|_{t=0}=u_0,\;\; \sum_{i,j=1}^n (\gamma a_{ij}(t,x,u)+(1-\gamma)\delta_{ij}) u_{x_j}\nu_i+ \gamma \psi(t,x,u)-(1-\gamma)\sum_{j=1}^n u_{0x_j}\nu_j |_{S}=0.\tag{73}\] For instance, consider the family 71 , 73 . As is easily seen, the conditions 12 13 are fulfilled for the functions \(\tilde{a}_{ij}=\gamma a_{ij}(t,x,u)+(1-\gamma)\delta_{ij}\) with the same constants and functions \(g_i\) and the constant \(\delta_0\) in 12 is replaced with \(\min(1,\delta_0)\). The new functions \(\tilde{a}_{ij}\), \(\tilde{\psi}=\gamma \psi(t,x,u)-(1-\gamma)\sum_{j=1}^n u_{0x_j}\nu_j\) also satisfy 14 16 and without loss of generality we can assume that the norms of all functions in these conditions are estimated by the constants independent of \(\gamma\). Thus, we can assume that the estimate \[\label{e48} \|u\|_{W_p^{1,2}(Q)}\leq C(M,M_1),\; \gamma\in [0,1],\tag{74}\] holds and the constant \(C(M,M_1)\) is independent of \(\gamma\in [0,1]\). Construct a function \(\Psi\in W_p^{1,2}(Q)\) such that \(\Psi|_{t=0}=u_0(x)\) and make the change of variables \(u=v+\Psi\). In order to construct this function, we can extend \(u_0\) to the whole \({\mathbb{R}}^n\) preserving the class (see Sect. 4.2.2, 4.2.3 in [13]) and find a solution to the Cauchy problem \(\Psi_t-\Delta \Psi=0\), \(\Psi|_{t=0}=u_0\) (see Theorem 5.7 in [20]). We arrive at the problem \[\begin{gather} \label{e49} M_0v= v_t-\sum_{i,j=1}^n \partial_{x_i}(\tilde{a}_{ij}(t,x,v+\Psi)v_{x_j})- \\ \sum_{i,j=1}^n \partial_{x_i}(\tilde{a}_{ij}(t,x,v+\Psi)\Psi_{x_j}) + {b}(x,t,v+\Psi,\nabla v+\Psi)+\Psi_t=0, \end{gather}\tag{75}\] \[\label{e50} v|_{t=0}=0,\;\; \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,v+\Psi) v_{x_j}+ \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,v+\Psi) \Psi_{x_j} + {\psi}(t,x,v+\Psi)|_{S}=0.\tag{76}\] Denote by \(\Phi(\gamma,w)\) a solution to the problem \[\begin{gather} \label{e51} M_0v= v_t-\sum_{i,j=1}^n \partial_{x_i}(\tilde{a}_{ij}(t,x,w+\Psi)v_{x_j})= \sum_{i,j=1}^n \partial_{x_i}( \tilde{a}_{ij}(t,x,w+\Psi)\Psi_{x_j}) \\ - {b}(x,t,w+\Psi,\nabla w+\Psi)-\Psi_t =f(w),\; \;v|_{t=0}=0,\; \end{gather}\tag{77}\] \[\label{e52} \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w+\Psi) v_{x_j}=- \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w+\Psi) \Psi_{x_j}- {\psi}(t,x,w+\Psi)|_{S}=g(w),\tag{78}\] where \(w\in H_1=\{w\in W_{p}^{1-\varepsilon_0,2-2\varepsilon_0}(Q): \; w(0,x)=0\}\) and \(\varepsilon_0\) is a positive number less than the minimum of the constants \(\varepsilon\) in ?? , ?? and the number \(1/2-(n+2)/4p\). Under this condition \(W_{p}^{1-\varepsilon_0,2-2\varepsilon_0}(Q)\subset W_{2p}^{1/2,1}\) (see Corollary 5.6.4 of Ch. 7 in [14]) or Theorem 1.22 in [18]). Moreover, we can assume decreasing \(\varepsilon_0\) if necessary that \(W_p^{1-\varepsilon_0,2-2\varepsilon_0}(Q)\subset B_{pq_4/(q_4-p) p}^{s_0+1/2p-1/2q+\varepsilon, 2s_0+1/p-1/q+2\varepsilon}(Q)\) (Corollary 5.6.4 of Ch. 7 in [14]) and thus we have the estimate \[\label{a52} \|v\|_{B_{pq_4/(q_4-p) p}^{s_0+\varepsilon, 2s_0+2\varepsilon}(S)}\leq \|v\|_{B_{pq_4/(q_4-p) p}^{s_0+1/2p-1/2q+\varepsilon, 2s_0+1/p-1/q+2\varepsilon}(Q)}\leq c_1\|v\|_{W_p^{1-\varepsilon_0,2-2\varepsilon_0}(Q)}\;\forall v\in H_1.\tag{79}\] Endow the space \(H_1\) with the norm coinciding with the norm in \(W_{p}^{1-\varepsilon_0,2-2\varepsilon_0}(Q)\). If \(w\in H_1\) then the right-hand sides in 77 , 78 belong to \(L_p(Q)\), \(W_p^{s_0,2s_0}(S)\), respectively, (see Lemma 2 and the proof of Theorem 5). Note that \[\begin{gather} \label{e53} W_{2p}^{1/2,1}(Q)\subset C^{1/2-(n+2)/4p, 1-(n+2)/2p}(\overline{Q}),\\ u\in W_{2p}^{1/2,1}(Q) \Rightarrow u|_{S}\in W_{2p}^{1/2-1/4p,1-1/2p}(S)\subset B_{q_3p}^{s_0}(0,T; L_{q_3}(\Gamma))\cap L_{q_3}(0,T; B_{q_3p}^{2s_0}(\Gamma)). \end{gather}\tag{80}\] By Theorem 1, there exists a unique solution to the problem 77 , 78 of the class \(W_p^{1,2}(Q)\). The following inequality is valid: \[\tilde{\delta}_0|\xi|^2\leq \sum_{i,j=1}^n \tilde{a}_{ij}\xi_i\xi_j \leq \frac{1}{\tilde{\delta}_0}|\xi|^2,\] where \(\tilde{\delta}_0=\min(\delta_0,\delta_1)\), \(1/\delta_1=\max(1/\delta_0,\sup_{|w|\leq R_0}\sup_{|\xi|=1}\sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w)\xi_i\xi_j)\), \(R_0=\|w\|_{C(\overline{Q})}\). In this case, we infer \[\label{e54} \|v\|_{W_{p}^{1,2}(Q)}\leq C_{0}(\|u_{0}\|_{W_{p}^{2-2/p}(G)}+ \|f(w)\|_{L_{p}(Q)}+\|g(w)\|_{W_{p}^{s_{0},2s_{0}}(S)}),\tag{81}\] where the constant \(C_0\) depends on \(\tilde{\delta}_0\) and the norms of \(\tilde{a}_{ij}\). Since the embedding \(W_{p}^{1,2}(Q)\subset W_{2p}^{1/2,1}(Q)\) [14] is compact, the estimate 81 implies that the mapping \(w\to \Phi(\gamma,w)\) is compact as well. Demonstrate that it is continuous. Let a sequence \(w_n\in H_1\) converges to \(w\in H_1\) in the norm of the space \(H_1\). Consider the sequence \(b_n=b(t,x,w_n+\Psi,\nabla (w_n+\Psi))\). In view of the condition 13 , we can conclude that \[\label{e55} \|b_n\|_{L_{p}(Q)}\leq c_1+ c_2(\|w_n\|_{L_{2p}(Q)}^2+ \|\nabla w_n\|_{L_{2p}(Q)}^2)\leq (c_3+c_4R_1),\; R_1=\max_n\|w_n\|_{H_1},\tag{82}\] Assign \(R_2=\max_n \|w_n+\Psi\|_{C^{1/2-(n+2)/4p, 1-(n+2)/2p}(\overline{Q})}\). In view of 80 , \(R_2<\infty\). By Lemma 3.1 of Ch. 2 in [1], there exists a subsequence \(w_{n_k}\) such that \(w_{n_k}\to w\), \(\nabla w_{n_k}\to \nabla w\) a.e. in \(Q\). By Theorem 4.9 in [21], choosing one more subsequence if necessary, we can say that there exists a function \(\psi\in L_{2p}(Q)\) such that \(|w_{n_k}|+|\nabla w_{n_k}|\leq \psi(t,x)\) a.e. In this case \(b_{n_k}-b\to 0\) a.e. and \(|b_{n_k}-b|^p\leq 2g_3^p+C_2 |\psi|^{2p}\) a. e. The Lebesgue dominated convergence theorem implies that \(\|b_{n_k}-b\|_{L_{p}(Q)}\to 0\) as \(k\to \infty\). Consider the second function \(l(w_n)= \sum_{i,j=1}^n \partial_{x_i}( \tilde{a}_{ij}(t,x,w_n+\Psi)\Psi_{x_j})\) occurring into \(f(w_n)\). We have that \[\begin{gather} l(w_n)= \sum_{i,j=1}^n (\tilde{a}_{ijx_i}(t,x,w_n+\Psi)\Psi_{x_j}+\tilde{a}_{iju}(t,x,w_n+\Psi)(w_n+\Psi)_{x_i}\Psi_{x_j}\\ +\tilde{a}_{ij}(t,x,w_n+\Psi)\Psi_{x_ix_j})=I_1(w_n)+I_2(w_n)+I_3(w_n). \end{gather}\] Since the functions \(\tilde{a}_{ij}\) are continuous and \(\|w_n-w\|_{C^{1/2-(n+2)/4p, 1-(n+2)/2p}(\overline{Q})}\to 0\) (see 80 ), \(\|I_3(w_n)-I_3(w)\|_{L_p(Q)}\to 0\) as \(n\to \infty\). Moreover, we have that \[\begin{gather} I_2(w_{n})-I_2(w)=\sum_{i,j=1}^n ( \frac{1}{2}(\tilde{a}_{iju}(t,x,w_n+\Psi)+\tilde{a}_{iju}(t,x,w+\Psi))(w_n-w)_{x_i}\Psi_{x_j}+\\ \frac{1}{2}(\tilde{a}_{iju}(t,x,w_n+\Psi)-\tilde{a}_{iju}(t,x,w+\Psi))(w_n+w)_{x_i}\Psi_{x_j}) \end{gather}\] This representation ensures the estimate \[\begin{gather} \|I_2(w_{n})-I_2(w)\|_{L_p(Q)}\leq c \|\nabla (w_n-w)\|_{L_{2p}(Q)}\|\nabla \Psi\|_{L_{2p}(Q)}+ \\ \sum_{i,j=1}^n \|(\tilde{a}_{iju}(t,x,w_n+\Psi)-\tilde{a}_{iju}(t,x,w+\Psi))\Psi_{x_j}\|_{L_{2p}(Q)}\|\nabla (w_n+w)\|_{L_{2p}(Q)}\to 0 \; \textrm{as}\;n\to \infty, \end{gather}\] where the last summand tends to 0 by the Lebesgue dominated convergence theorem. At last, we derive that \[\begin{gather} \|I_1(w_{n_k})-I_1(w)\|_{L_p(Q)}\leq c\sum_{i,j=1}^n \|(\tilde{a}_{ijx_i}(t,x,w_{n_k}+\Psi)-\tilde{a}_{ijx_i}(t,x,w+\Psi))\Psi_{x_j}\|_{L_{p}(Q)}\leq \\ C\sum_{i,j=1}^n \|\tilde{a}_{ijx_i}(t,x,w_{n_k}+\Psi)-\tilde{a}_{ijx_i}(t,x,w+\Psi)\|_{L_{q_3}(Q)}\to 0 \; \textrm{as}\;k\to \infty \end{gather}\] also in view of the Lebesgue dominated convergence theorem. Indeed, \(\tilde{a}_{ijx_i}(t,x,w_{n_k}+\Psi)-\tilde{a}_{ijx_i}(t,x,w+\Psi))\to 0\) a.e. in \(Q\). On the other hand, \(|\tilde{a}_{ijx_i}(t,x,w_{n_k}+\Psi)-\tilde{a}_{ijx_i}(t,x,w+\Psi)|\leq 2\|\tilde{a}_{ijx_i}(t,x,w)\|_{C([-R_2,R_2])}\) and the last function is integrable in view of ?? . Finally we can say that \(\|f(w_{n_k})-f(w)\|_{L_p(Q)}\to 0\) as \(k\to \infty\). Now, we study the question of convergence of the function \(g(w_n) = - \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w_n+\Psi) \Psi_{x_j}- {\psi}(t,x,w_n+\Psi)|_{S}=J_1(w_n)+J_2(w_n).\) Estimate \(\|J_2(w_{n_k})-J_2(w)\|_{W_p^{s_0,2s_0}(S)}.\) It suffices to establish the necessary estimate in the simplest case \(S=(0,\infty)\times \Gamma\), \(\Gamma={\mathbb{R}}^{n-1}\), \(Q=(0,\infty)\times {\mathbb{R}}^n_+\), \({\mathbb{R}}^n_+=\{x\in {\mathbb{R}}^n: \; x_n>0\}\) assuming that all functions are compactly supported. The norm in this case is defined by the equality 38 . We have that \(\Delta_{\tau,t}(J_2(w_{n_k})-J_2(w))=[\psi(t+\tau,x,w_{n_k}(t+\tau,x)+\Psi(t+\tau,x))- \psi(t,x,w_{n_k}(t+\tau,x)+\Psi(t+\tau,x))-\psi(t+\tau,x,w(t+\tau,x)+\Psi(t+\tau,x))+ \psi(t,x,w(t+\tau,x)+\Psi(t+\tau,x))]+[\psi (t,x, w_{n_k}(t+\tau,x)+\Psi(t+\tau,x))-\psi(t,x, w_{n_k}(t,x)+\Psi(t,x))-\psi (t,x, w(t+\tau,x)+\Psi(t+\tau,x))+\psi(t,x,w(t,x)+\Psi(t,x))] =I_{1k}+I_{2k}\). The expression \(\Delta_{h,x}(J_2(w_{n_k})-J_2(w))\) can be represented similarly. The triangle inequality yields \[\begin{gather} \int_{0}^{\delta} \frac{1}{|\tau|^{1+s_0p}} \|\Delta_{\tau,t} (J_2(w_{n_k})-J_2(w))\|_{L_{p}(S)}^p\,d\tau\leq c(p)\Bigl( \int_{0}^{\delta} \frac{1}{|\tau|^{1+s_0p}} \|I_{1k}(\tau,t,x)\|_{L_{p}(S)}^p\,d\tau+\\ \int_{0}^{\delta} \frac{1}{|\tau|^{1+s_0p}} \|I_{2k}(\tau,t,x)\|_{L_{p}(S)}^p\,d\tau\Bigr)=c(p)(J_{1k}+J_{2k}). \end{gather}\] We have that \(I_{1k}\to 0\) as \(k\to\infty\) a.e. on \(S\times (0,\delta)\) and \(|I_1|\leq c\|\Delta_{\tau,t}\psi(t,x,w)\|_{C([-R_2,R_2])}\) and the right-hand side here to the power \(p\) with the weight \(1/|\tau|^{1+s_0p}\) is integrable. The Lebesgue dominated converge theorem ensures that \(J_{1k}\to 0\) as \(k\to \infty\). Next, we derive that \[\begin{gather} I_{2k}(\tau,t,x)=\int_0^1 \psi_u(t,x, w_{n_k}(t,x)+\Psi(t,x)+r(w_{n_k}(t+\tau,x)-w_{n_k}(t,x)+\Psi(t+\tau,x)- \\ \Psi(t,x)))dr (w_{n_k}(t+\tau,x)-w_{n_k}(t,x)+\Psi(t+\tau,x)-\Psi(t,x))- \int_0^1 \psi_u(t,x, w(t,x)+\Psi(t,x)+\\ r(w(t+\tau,x)-w(t,x)+ \Psi(t+\tau,x)-\Psi(t,x)))dr (w(t+\tau,x)-w(t,x)+\Psi(t+\tau,x)-\Psi(t,x)). \end{gather}\] This equality can be rewritten in the form \[\begin{gather} I_{2k}(\tau,t,x)=\frac{1}{2}\int_0^1 \psi_u(t,x, w_{n_k}(t,x)+\Psi(t,x)+r(\Delta_{\tau,t}(w_{n_k}(t,x)+\Psi(t,x)))) + \\ \psi_u(t,x, w(t,x)+ \Psi(t,x)+ r(\Delta_{\tau,t}(w(t,x)+ \Psi(t,x)))) dr \Delta_{\tau,t}(w_{n_k}(t,x)-w(t,x))+ \\ \frac{1}{2}\int\limits_0^1 \psi_u(t,x, w_{n_k}(t,x)+ \Psi(t,x)+r(\Delta_{\tau,t}(w_{n_k}(t,x)+\Psi(t,x)))) - \psi_u(t,x, w(t,x)+\Psi(t,x) \\ + r(\Delta_{\tau,t}(w(t,x)+ \Psi(t,x)))) dr \Delta_{\tau,t}(w_{n_k}(t,x)+w(t,x)+ 2\Psi(t,x))=\\ \frac{1}{2}\int\limits_0^1 \tilde{I}_{1k}(\tau,t,x,r)\,dr \Delta_{\tau,t}(w_{n_k}(t,x)-w(t,x)) + \frac{1}{2}\int\limits_0^1 \tilde{I}_{2k}(\tau,t,x,r)\,dr \Delta_{\tau,t}(w_{n_k}(t,x)+\\ w(t,x)+ 2\Psi(t,x)) = \frac{1}{2}\int\limits_0^1 (\tilde{I}_{1k}(\tau,t,x,r)+\tilde{I}_{2k}(\tau,t,x,r))\,dr \Delta_{\tau,t}(w_{n_k}(t,x)-w(t,x)) + \\ \frac{1}{2}\int\limits_0^1 \tilde{I}_{2k}(\tau,t,x,r)\,dr \Delta_{\tau,t}(2w(t,x)+ 2\Psi(t,x)). \end{gather}\] In view of 79 , \[\label{a53} \|w_{n_k}-w\|_{B_{pq_4/(q_4-p) p}^{s_0}(S)}\leq c \|w_{n_k}-w\|_{H_1} \to 0\;\textrm{as}\; k\to \infty.\tag{83}\] The Hölder inequality implies that \[\begin{gather} J_{2k}\leq c \|\psi_u (t,x,w)\|_{L_{q_4}(S;C([-R_2,R_2]))}\|w_{n_k}-w\|_{B_{pq_4/(q_4-p) p}^{s_0}(S)}+ \\ c_1\int_0^\delta \frac{1}{\tau^{1+s_0p}}\Bigl( \int_0^1\int_S |\tilde{I}_{2k}|^{q_4}\,dSdr\Bigr)^{p/q_4} \|\Delta_{\tau,t}(w+\Psi)\|_{L_{pq_4/(q_4-p) p}(S)}^p\,d\tau. \end{gather}\] The former summand on the right-hand side tend to zero as \(k\to \infty\) in view of 83 . The Lebesgue dominated converge theorem ensures that \(\int_0^1\int_S |\tilde{I}_{2k}|^{q_4}\,dSdr\to 0\) as \(k\to \infty\). Indeed, \(|\tilde{I}_{2k}|^{q_4}\leq \|\psi_u (t,x,w)\|^{q_4}_{C([-R_2,R_2])}\) and the last function is integrable. Moreover, for all \(r\in (0,1), \tau>0\) and almost all \((t,x)\) \(I_{2k}\to 0\) as \(k\to \infty\). Again, the Lebesgue dominated convergence theorem and the estimate \[\begin{gather} \Bigl( \int_0^1\int_S |\tilde{I}_{2k}|^{q_4}\,dSdr\Bigr)^{p/q_4} \|\Delta_{\tau,t}(w+\Psi)\|_{L_{pq_4/(q_4-p) p}(S)}^p\leq \\ c_2 \|\psi_u (t,x,w)\|_{L_{q_4}(Q;C^([-R_2,R_2]))}^{p} \|\Delta_{\tau,t}(w+\Psi)\|_{L_{pq_4/(q_4-p) p}(S)}^p \end{gather}\] ensures that the second integral tend to 0 as \(k\to \infty\). Thus, \(J_{2k}\to 0\) as \(k\to \infty\). We have proven that \(J_2(w_{n_k})-J_2(w)\to 0\) as \(k\to \infty\) in \(W_p^{s_0}(0,T;L_p(\Gamma))\). Similar arguments prove that \(J_2(w_{n_k})-J_2(w)\to 0\) as \(k\to \infty\) in \(L_p(0,T;W_p^{2s_0}(\Gamma))\). Finally, \(\|\psi(t,x,w_{n_k}+\Psi) - \psi(t,x,w+\Psi)\|_{W_p^{s_0,2s_0}(S)}\to 0\) as \(k\to \infty\). Similar arguments are employed in the proof of the convergence \(\sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w_n+\Psi) \Psi_{x_j}\to \sum_{i,j=1}^n \tilde{a}_{ij}(t,x,w+\Psi) \Psi_{x_j}\) in the space \(W_p^{s_0,2s_0}(S)\).

Denote by \(v_n,v\) the solutions to the problem 77 , 78 relating to the \(w_n,w\). In this case the difference \(\varphi_{k}=v_{n_k}-v\) is a solution to the problem \[\begin{gather} \label{e56} M_0\varphi_n= \varphi_{kt}-\frac{1}{2}\sum_{i,j=1}^n \partial_{x_i}((\tilde{a}_{ij}(t,x,w_{n_k}+\Psi)+ \tilde{a}_{ij}(t,x,w+\Psi))\varphi_{kx_j})= \\ \frac{1}{2}\sum_{i,j=1}^n \partial_{x_i}((\tilde{a}_{ij}(t,x,w_{n_k}+\Psi) - \tilde{a}_{ij}(t,x,w+\Psi))(v_{nx_j}+v_{x_j})) + f(w_n)-f(w), \end{gather}\tag{84}\] The right-hand side can be written as \(\tilde{f}(w_{n_k})-\tilde{f}(w)\). \[\begin{gather} \label{e57} \varphi_k|_{t=0}=0,\;\; \sum_{i,j=1}^n \frac{1}{2} (\tilde{a}_{ij}(t,x,w_{n_k}+\Psi)+ \tilde{a}_{ij}(t,x,w+\Psi)) \varphi_{kx_j}= - \sum_{i,j=1}^n \frac{1}{2} (\tilde{a}_{ij}(t,x,w_{n_k}+\Psi)\\ - \tilde{a}_{ij}(t,x,w+\Psi)) (v_{n_kx_j}+v_{x_j})+ g(w_n)-g(w)=\tilde{g}(w_n)-\tilde{g}(w) . \end{gather}\tag{85}\] The functions \(v_n\) meet the estimate 81 and, hence, the norms \(\|v_n\|_{W_p^{1,2}(Q)}\) are bounded uniformly in \(n\). In this case the difference \(\varphi_k\) also satisfies the estimate \(\eqref{e54}\) with the right-hand side \(c(\|\tilde{f}(w_{n_k})-\tilde{f}(w)\|_{L_p(Q)}+ \|\tilde{g}(w_{n_k})-\tilde{g}(w)\|_{W_p^{s_0,2s_0}(S)})\). As in the proofs of the convergence \(f(w_{n_k})\to f(w)\), \(g(w_{n_k})\to g(w)\), we can demonstrate that \(\tilde{f}(w_{n_k})\to \tilde{f}(w)\), \(\tilde{g}(w_{n_k})\to \tilde{g}(w)\). Thus, the convergence \(w_n\to w\) in \(H_1\) implies that there exists a subsequence \(w_{n_k}\) such that \(\Phi(\gamma, w_{n_k})\to \Phi(\gamma, w)\). In this case \(\Phi(\gamma, w_{n})\to \Phi(\gamma, w)\) as \(n\to \infty.\) We have proven that the mapping \(w\to \Phi(\gamma, w)\) is continuous and compact, its continuity in the parameter \(\gamma\in [0,1]\) is obvious.

As is easily seen, \(v\) is a solution to the problem 77 , 78 if and only if \(v\) is a fixed point of the mapping \(\Phi(\gamma,w)\), i. e. \[\label{e58} v=\Phi(\gamma,v).\tag{86}\] For \(\gamma=0\), a solution to the problem 73 , 74 exists and \(v+\Psi=u\) is a solution to the problem \(u_t-\Delta u=0\), \(u|_{t=0}=u_0\), \(\frac{\partial u}{\partial \nu}=\frac{\partial u_0}{\partial \nu}\). Moreover, we have estimates uniform on the parameter \(\gamma\). By Theorem 37.6 in [22] the equation 86 has a solution for all \(\gamma\in [0,1]\). Now we prove uniqueness of solutions to our problem 1 , 3 . Let \(u_1,u_2\) be two solutions of the problem. Subtracting equations 1 for \(u_1\) and \(u_2\), multiplying the inequality obtained by \(v=u_1-u_2\), and integrating over \(G\), we derive that \[\begin{gather} \label{e59} \partial_t \int_G \frac{v^2}{2} + \frac{1}{2}\sum_{i,j=1}^n (a_{ij}(t,x,u_1)+ a_{ij}(t,x,u_2)) v_{x_j}v_{x_i}\,dx =-\int_{\Gamma}(\psi(t,x,u_1)-\psi(t,x,u_2))v\,d\Gamma \\ + \int_G\frac{1}{2}\sum_{i,j=1}^n ({a}_{ij}(t,x,u_1)- \tilde{a}_{ij}(t,x,u_2))(u_{1x_j}+u_{2x_j})v_{x_i} -(b(t,x,u_1)-b(t,x,u_2))v\,dx \end{gather}\tag{87}\] Fix \(t\in (0,T)\). Integrating from 0 to \(\tau\leq t\) and using the conditions on the coefficients, we arrive at the inequality \[\begin{gather} \label{e60} \int_G \frac{v^2}{2}(\tau,x) \,dx+ \int_0^\tau\int_G\delta_0|\nabla v(\xi,x)|^2\,dxd\xi \leq \int_0^\tau \int_{\Gamma}|(\psi(\xi,x,u_1)-\psi(\xi,x,u_2))v|\,d\Gamma d\xi \\ + c_1\int_0^\tau\int_G|\nabla (u_{1}+u_{2})| |v||\nabla v| +|(b(\xi,x,u_1)-b(\xi,x,u_2))v|\,dGd\xi\leq \\ \int_0^t \int_{\Gamma}|(\psi(\xi,x,u_1)-\psi(\xi,x,u_2))v|\,d\Gamma d\xi \\ + c_1\int_0^t\int_G|\nabla (u_{1}+u_{2})| |v||\nabla v| +|(b(\xi,x,u_1,\nabla u_1)-b(\xi,x,u_2,\nabla u_2))v|\,dGd\xi. \end{gather}\tag{88}\] Taking the maximum in \(\tau\) on the left-hand side, we conclude that \[\begin{gather} \label{e61} \max_{\tau\in [0,t]}\int_G \frac{v^2(\tau,x)}{2} \,dx+ \int_0^t\int_G\delta_0|\nabla v|^2\,dxd\xi\leq \int_0^t \int_{\Gamma}|(\psi(\xi,x,u_1)-\psi(\xi,x,u_2))v|\,d\Gamma d\xi \\ + c_1\int_0^t\int_G|\nabla (u_{1}+u_{2})| |v||\nabla v| +|(b(\xi,x,u_1,\nabla u_1)-b(\xi,x,u_2,\nabla u_2))v|\,dGd\xi. \end{gather}\tag{89}\] Estimate every summand on the right-hand side. Let \(R=\|u_1\|_{C(S)}+\|u_2\|_{C(S)}\). We have \[\begin{gather} \label{e62} \int\limits_0^t \int\limits_{\Gamma}|(\psi(\xi,x,u_1)-\psi(\xi,x,u_2))v|\,d\Gamma d\xi= \int\limits_0^t \int\limits_{\Gamma}\Bigl|\int_0^1\psi_u(\xi,x,u_1+s(u_2-u_1))\,ds\Bigr| |v|^2\,d\Gamma d\xi \\ \leq \int\limits_0^t \|\psi(\xi,x,u)\|_{L_{q_4}(\Gamma;C^([-R,R]))}\|v\|_{L_{2 q_4'}(\Gamma)}^2\, d\xi. \end{gather}\tag{90}\] The embedding theorems [18] and the trace theorems [18] validate the estimate \[\label{e63} \|v\|_{L_{2 q_4'}(\Gamma)}^2\leq c \|v\|_{W_{2}^s(\Gamma)}^2\leq \|v\|_{W_{2}^{s+1/2}(G)}^2\leq c_2 \|v\|_{W_{2}^{1}(G)}^{2(s+1/2)}\|v\|_{L_{2}(G)}^{2(s-1/2)},\tag{91}\] where \(s=(n-1)/2q_4\). Note that \(s+1/2<1\). In this case the inequality 23 yields \(\|v\|_{L_{2 q_4'}(\Gamma)}^2\leq \varepsilon \|v\|_{W_{2}^{1}(G)}^{2}+c(\varepsilon) \|v\|_{L_{2}(G)}^{2}\). This inequality and 90 imply that \[\label{e65} \int_0^t \int_{\Gamma}|(\psi(\xi,x,u_1)-\psi(\xi,x,u_2))v|\,d\Gamma d\xi\leq \varepsilon \int_0^t \|\nabla v\|_{L_2(G)}^2\,d\xi + c(\varepsilon) t \|v\|_{L_{\infty}(0,t;L_2(G))}^{2}.\tag{92}\] Estimate the second summand in 89 . The Hölder inequality ensures that \[\label{e66} \int_0^t\int_G|\nabla (u_{1}+u_{2})| |v||\nabla v|\, dxd\xi \leq \int_0^t\|\nabla (u_{1}+u_{2})\|_{L_q(G)} \|\nabla v\|_{L_2(G)}\| v\|_{L_{2q/(q-2)}(G)}\,d\xi.\tag{93}\] Choose \(q=np/(n+2-p)\). In this case (see Theorem 4.10.2 of Ch. III in [8] or Theorem 1.23 in [18]) \(\|\nabla (u_{1}+u_{2})\|_{L_q(G)}\leq c \|\nabla (u_{1}+u_{2})(\xi,x))\|_{W_p^{1-2/p}(G)}\leq c_1\|u_{1}+u_{2}\|_{W_p^{1,2}(Q)}\). As before, \[\| v\|_{L_{2q/(q-2)}(G)}\leq c\|v\|_{W_{2}^{1}(G)}^{2s}\|v\|_{L_{2}(G)}^{2(1-s)}, \; s=n/q<1.\] In accord with 93 , 23 , we infer \[\begin{gather} \label{e671} \int_0^t\int_G|\nabla (u_{1}+u_{2})| |v||\nabla v\|\, dx \leq c_2\int_0^{t}\| \nabla v\|_{L_2(G)}^{1+s}\| v\|_{L_{2q/(q-2)}(G)}^{1-s}\,d\xi\leq \\ \varepsilon \int_0^t \|\nabla v\|_{L_2(G)}^2\,d\xi + c(\varepsilon) t \|v\|_{L_{\infty}(0,t;L_2(G))}^{2}, \end{gather}\tag{94}\] where \(\varepsilon>0\) is arbitrary. Now take \(p>n+2\). The condition 67 implies that \[\begin{gather} \label{e67} \int_0^t\int_G|(b(\xi,x,u_1,\nabla u_1)-b(\xi,x,u_2,\nabla u_2))v|\,dGd\xi\leq \\ \int_0^t\int_G g_1(\xi,x)|v|^2 + g_2(\xi,x) |\nabla v| |v| \,dxd\xi. \end{gather}\tag{95}\] For the former summand we have the estimate \[\begin{gather} \label{e68} \int_0^t\int_G g_1 |v|^2\,d\xi\leq \int_0^t \|g_1\|_{L_{q_6}(G)} \|v\|_{L_{2q_6'}(G)}^2\,d\xi \leq \int_0^t \|g_1\|_{L_{q_6}(G)} \|v\|_{W_{2}^1(G)}^{2s}\|v\|_{L_{2}(G)}^{2(1-s)}\,d\xi \leq \\ \varepsilon \int_0^t \|\nabla v\|_{L_{2}(G)}^{2}\,d\xi + c(\varepsilon)\|v\|_{L_{\infty}(0,t;L_2(G))}^{2}(c(\varepsilon)\int_0^t \|g_1\|_{L_{q_6}(G)}^{1/(1-s)}\,d\xi+\varepsilon t),\;s=n/2q_6, \end{gather}\tag{96}\] where we employ the equality \(\|v\|_{W_2^1(G)}^2=\|\nabla v\|_{L_2(G)}^2+\| v\|_{L_2(G)}^2\). Note that \(1/(1-s)\leq q_6\). In view of the absolute continuity of the integral, \(\int_0^t \|g_1\|_{L_{q_6}(G)}^{1/(1-s)}\,d\xi\to 0\) as \(t\to 0\). The integral \(\int_0^t\int_G g_2(\xi,x) |\nabla v| |v| \,dxd\xi\) is estimated similarly. We have \[\begin{gather} \label{e69} \int_0^t\int_G g_2(\xi,x) |\nabla v| |v| \,dxd\xi\leq \int_0^t \|g_2\|_{L_{q_7}(G)} \|\nabla v\|_{L_{2}(G)}\| v\|_{L_{2q_7'/(2-q_7')}(G)}\,d\xi \leq \\ \int_0^t \|g_2\|_{L_{q_7}(G)} \|v\|_{W_{2}^1(G)}^{1+s}\|v\|_{L_{2}(G)}^{1-s}\,d\xi \leq \varepsilon \int_0^t \|\nabla v\|_{L_{2}(G)}^{2}\,d\xi+ \\ \|v\|_{L_{\infty}(0,t;L_2(G)}^{2}(c(\varepsilon)\int_0^t \|g_2\|_{L_{q_7}(G)}^{2/(1-s)}\,d\xi+\varepsilon t),\;s=n/q_7. \end{gather}\tag{97}\] Note that \(2/(1-s)\leq q_7\). In this case, the relations 92 , 94 , 95 , 96 , 97 and 89 , ensure the estimate \[\begin{gather} \label{e70} \max_{\tau\in [0,t]}\int\limits_G \frac{v^2(\tau,x)}{2} \,dx+ \int\limits_0^t\int_G\delta_0|\nabla v|^2\,dxd\xi\leq 4\varepsilon \int\limits_0^t \|\nabla v\|_{L_{2}(G)}^{2}\,d\xi+ c(\varepsilon)\|v\|_{L_{\infty}(0,t;L_2(G))}^{2}\varphi(t), \end{gather}\tag{98}\] where \(\varphi(t)\) is a continuous function such that \(\varphi(0)=0\). Choose \(\varepsilon\) so that \(4\varepsilon<\delta_0/2\) and \(t_0\) so that \(c(\varepsilon)\varphi(t)<1/4\) for \(t\leq t_0\). In this case the inequality \(\eqref{e70}\) implies that \(v(t,x)=0\) for \(t\leq t_0\). Repeating the arguments for \(t\geq t_0\), we obtain that \(v=0\) on some segment \([t_0,t_1]\), and so on. It is easy to see that the length of the segments \([t_{i-1},t_i]\) does not tend to 0, since it depends on the properties of the functions \(g_i\) and constants arising in the inequalities are really constants from the embedding theorems and interpolaltion inequalities. Proceed with the case of \(p\in ((n+2)/2,n+2]\). In this case we have an estimate \[\begin{gather} \label{e71} \int_0^t\int_G|(b(\xi,x,u_1,\nabla u_1)-b(\xi,x,u_2,\nabla u_2))v|\,dGd\xi\leq \\ \int_0^t\int_G g_1 v^2 (1+(|\nabla u_1|+\nabla u_2|)^\alpha) +g_2|\nabla v| |v| (1+ (|\nabla u_1|+\nabla u_2|)^\beta)\,dxd\xi. \end{gather}\tag{99}\] The summands of the form \(\int_0^t\int_G g_1 v^2 + g_2|\nabla v| |v|\,dxd\xi\) have been already estimated. Estimate the expressions \(\int_0^t\int_G g_1 v^2(|\nabla u_1|+\nabla u_2|)^\alpha +g_2|\nabla v| |v| (|\nabla u_1|+\nabla u_2|)^\beta\,dxd\xi\). As in the proof of 96 , we have \[\begin{gather} \label{e72} I_1=\int_0^t\int_G g_1 v^2(|\nabla u_1|+\nabla u_2|)^\alpha \,dxd\xi\leq \varepsilon \int_0^t \|\nabla v\|_{L_{2}(G)}^{2}\,d\xi+\\ \|v\|_{L_{\infty}(0,t;L_2(G)}^{2}(c(\varepsilon)\int_0^t \|g_1(|\nabla u_1|+|\nabla u_2|)^\alpha\|_{L_{q}(G)}^{1/(1-s)}\,d\xi +\varepsilon t),\;s=n/2q, \end{gather}\tag{100}\] where \(q=q_8np/(np+\alpha q_8(n+2-p))< q_8\) for \(p<n+2\). For \(p=n+2\), we take \(nq_8/2(q_8-1)\geq q<q_8\). Let \(p<n+2\). The Hölder inequality yields \[\begin{gather} \label{e73} \int_0^t \|g_1(|\nabla u_1|+|\nabla u_2)^\alpha|\|_{L_{q}(G)}^{1/(1-s)}\,d\xi \leq \int_0^t \|g_1\|_{L_{q_8}(G)}^{1/(1-s)} \||\nabla u_1|+|\nabla u_2|\|_{L_{\alpha qq_8/(q_8-q)}(G)}^{\alpha/(1-s)}\,d\xi\leq \\ c \int_0^t \|g_1\|_{L_{q_8}(G)}^{1/(1-s)}\,d\xi, \end{gather}\tag{101}\] where \(\alpha qq_8/(q_8-q)=np/(n+2-p)\) and, thereby, \(\max_t\||\nabla u_1|+|\nabla u_2|\|_{L_{\alpha qq_8/(q_8-q)}(G)}\leq c(\|u_1\|_{W_p^{1,2}(Q)}+\|u_2\|_{W_p^{1,2}(Q)})\). Let \(p=n+2\). Again, we have that \[\begin{gather} \label{e74} \int_0^t \|g_1(|\nabla u_1|+|\nabla u_2)^\alpha|\|_{L_{q}(G)}^{1/(1-s)}\,d\xi \leq \int_0^t \|g_1\|_{L_{q_8}(G)}^{1/(1-s)} \||\nabla u_1|+|\nabla u_2|\|_{L_{\alpha qq_8/(q_8-q)}(G)}^{\alpha/(1-s)}\,d\xi\leq\\ c \int_0^t \|g_1\|_{L_{q_8}(G)}^{1/(1-s)}\,d\xi. \end{gather}\tag{102}\] We have used the fact that, for \(p=n+2\), \(\nabla u\in C([0,T];L_q(G))\) for every \(q\geq 1\). The condition ov the parameter \(q_8\) in 68 ensures the inequality \(1/(1-s)\leq q_8\). As before, we infer \[\begin{gather} \label{e75} \int_0^t\int_G g_2 |v| |\nabla v| (|\nabla u_1|+\nabla u_2|)^\beta)\,dxd\xi\leq \varepsilon \int_0^t \|\nabla v\|_{L_{2}(G)}^{2}\,d\xi+ \\ \|v\|_{L_{\infty}(0,t;L_2(G)}^{2}(c(\varepsilon)\int_0^t \|g_2(|\nabla u_1|+|\nabla u_2|)^\beta\|_{L_{q}(G)}^{2/(1-s)}\,d\xi+\varepsilon t),\;s=n/q, \end{gather}\tag{103}\] where \(q=q_9np/(np+\beta q_9(n+2-p))\) for \(p<n+2\) and \(q\in (nq_9/(q_9-1),q_9)\) for \(p=n+2\). Let us consider the case of \(p<n+2\). We can conclude that \[\begin{gather} \label{e76} \int_0^t \|g_2(|\nabla u_1|+|\nabla u_2)^\beta|\|_{L_{q}(G)}^{2/(1-s)}\,d\xi \leq \\ \int_0^t \|g_1\|_{L_{q_9}(G)}^{2/(1-s)} \||\nabla u_1|+|\nabla u_2|\|_{L_{\beta qq_9/(q_9-q)}(G)}^{2\beta/(1-s)}\,d\xi\leq c \int_0^t \|g_1\|_{L_{q_8}(G)}^{2/(1-s)}d\xi, \end{gather}\tag{104}\] where \(\beta qq_9/(q_9-q)=np/(n+2-p)\) and, thus, the quantity \[\max_t\||\nabla u_1|+|\nabla u_2|\|_{L_{2\beta qq_9/(q_9-q)}(G)}\leq c(\|u_1\|_{W_p^{1,2}(Q)}+\|u_2\|_{W_p^{1,2}(Q)})\] is finite. The same estimate holds for \(p=n+2\). The conditions on \(q_9\) in 68 ensures the inequality \(2/(1-s)\leq q_9\). The inequalities 100 104 imply that the inequality of the form 98 holds which ensures uniqueness of solutions. The proofs in the case of the problem 1 , 2 are much simpler and the sequence of arguments is the same. So we omit the proofs in this case. ◻

Proof. Denote \(R=\|\nabla u_0\|_{C(\overline{G})}\). Construct functions \(\varphi,\psi\) such that \(\varphi(u)=u\) for \(u\in [-m_0+m,M+m_0]\), \(\varphi(u)=M+m_0\) for \(u\geq M+m_0\), \(\varphi(u)=-m_0+m\) for \(u\geq -m_0+m\), \(\psi(\vec{p})=\vec{p}\) for \(|\vec{p}|\leq 2R\), \(\psi(\vec{p})=0\) for \(|\vec{p}|>3R\), \(\psi\in C^1({\mathbb{R}}^n)\). Define the function \(\tilde{b}(t,x,u,\vec{p})=b(t,x,\varphi(u),\psi(\vec{p}))\). Consider the problem 1 , 2 or the problem 1 , 3 , where the function \(b(t,x,u,\vec{p})\) is replaced with \(\tilde{b}(t,x,u,\vec{p})\). As is easily seen, the conditions of Theorem 7 are fufilled. In this case there exists a solution \(u\in W_p^{1,2}(Q)\) to the problem 1 , 2 or, respectively, to the problem 1 , 3 . Since \(W_p^{1,2}(Q)\subset C^{1-(n+2)/2p, 2-(n+2)/p}(\overline{Q})\), there exists \(\tau_0>0\) such that \(u(t,x)\in [-m_0+m,M+m_0]\), \(\nabla u(t,x)\in B_{2R}\) in \(Q_{\tau_0}\). Then \(\varphi(u)=u\), \(\psi(\nabla u)=\nabla u\) in \(Q_{\tau_0}\). Hence, the function \(u\) is a solution to the problem 1 , 2 or, respectively, 1 , 3 in \(Q_{\tau_0}\). ◻