On semilinear damped wave equations with initial data in homogeneous Sobolev spaces
January 01, 1970
Abstract
In this paper, we study semilinear damped equations \(u_{tt}+u_t-\Delta u=|u|^p\) with the initial data in \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) with the dimension \(n\le n\). Chen-Reissig [1] studied the case \(0<\gamma\le\min\{\frac{n}{2}, (-n+\sqrt{n^2+16n})/4\}\) and showed that the exponent \(p_{\mathrm{crit}}=1+4/(n+2\gamma)\) of \(p\) distinguishes the time global existence and the blow-up of solution. In this paper, we discuss the case \(\gamma\ge\min\{\frac{n}{2}, (-n+\sqrt{n^2+16n})/4\}\) and show that the critical exponent is not \(1+4/(n+2\gamma)\) but \[\begin{align} \begin{cases} 1+\frac{2}{n} & \text{if }n=1\text{ or }2\\ \frac{n+\sqrt{n^2+16n}}{2n} & \text{if }3\le n\le 6. \end{cases} \end{align}\]
1 Introduction↩︎
In this paper, we study the following Cauchy problem for the semilinear damped wave equation: \[\begin{align} \begin{cases} \partial^2_tu+\partial_tu-\Delta u=|u|^p&x\in\mathbb{R}^n, t>0,\\ \left(u(0,x),\partial_tu(t,x)|_{t=0}\right)=(\epsilon u_0,\epsilon u_1)&x\in\mathbb{R}^n, \end{cases}\label{bdw} \end{align}\tag{1}\] with \(p>1\) and \(0<\epsilon\ll1\). In particular, we study the case when \((u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2), \gamma>0\) and \(s\in(0,1]\). Here, \[\begin{align} \|f\|_{{\dot{H}^{-\gamma}}}:=\left\||\xi|^{-\gamma}\hat{f}\right\|_{L^2}. \end{align}\] Our main purpose is to determine the critical exponent \(p_\mathrm{c}\) of the power \(p\) in 1 which distinguishes the global existence of solutions and their blow-up, and the lifespan \(T\) when blow-up.
With initial data \((u_0,u_1)\in (L^1\cap H^s)\times (L^1\cap L^2)\), many papers (for example, [2]–[8]) show that the critical exponent of 1 is the Fujita exponent \(p_{\mathrm{F}} := 1+\frac{2}{n}\), which is also the critical exponent of the semilinear heat equation (see [9]).
However under different class of initial data, the Cauchy problem 1 admits a different critical exponent. For example, when the initial data satisfies \((u_0,u_1) \in(L^m\cap H^s)\times(L^m\cap L^2)\) with \(0<s\le1\) and \((-n+\sqrt{n^2+16n})/4<m<2\), the critical exponent is \[\begin{align} 1+\frac{2m}{n}\label{pcm} \end{align}\tag{2}\] as demonstrated in [10], [11].
Recently, the critical exponent of the Cauchy problem 1 with initial data \((u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) has been studied in light of the Hardy-Littlewood-Sobolev inequality: \(L^m\subset{\dot{H}^{-\gamma}}\) for \(1<m\le2\), \(0\le \gamma<n/2\) and \(1/m=1/2+\gamma/n\). In the case \(1\le n\le6\), Chen and Reissig [1] established that for \((u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) with \(0\le\gamma<\min\{n/2,\tilde{\gamma}\}\) here \(\tilde{\gamma}:=\frac{-n+\sqrt{n^2+16n}}{4}\), the critical exponent for the Cauchy problem 1 is given by \[\begin{align} p_{\mathrm{crit}}:=1+\frac{4}{n+2\gamma}\label{pcga} \end{align}\tag{3}\] (See Figure 1). Then, we remark that \(p_{\mathrm{crit}}\) in 3 with \(1/m=1/2+\gamma/n\), which appears as the condition of Hardy-Littlewood-Sobolev inequality, coinsides with the index in 2 . Moreover, D’Abbicco [12] proved the existence of global solutions in the critical case. Furthermore, [1] derived sharp lifespan estimates for blow-up weak solutions: if \(p<p_{\mathrm{crit}}\), then the lifespan satisfies \[\begin{align} T\lesssim\epsilon^{-1/(p'-1-n/4-\gamma/2)}=\epsilon^{-1/(p'-p'_\mathrm{crit})}. \end{align}\] In particular, if \(p>1+2\gamma/n\) in addition to \(p<p_{\mathrm{crit}}\), then the estimate is sharp \[\begin{align} T\simeq \epsilon^{-1/(p'-1-n/4-\gamma/2)}=\epsilon^{-1/(p'-p'_\mathrm{crit})}. \end{align}\]
Figure 1: Description of the critical exponent in the \(\gamma-p\) plane (with \(s=1\)).
In this paper, we studied the Cauchy problem 1 when the initial data in \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) under conditions that could not be handled in [1], that is, the case \(\gamma\ge \min\{n/2,\tilde{\gamma}\}\), and we obtain new critical exponents not 3 but \[\begin{align} \begin{cases} 1+\frac{2}{n}&\text{ if }n=1\text{ or }2,\\ \frac{n+\sqrt{n^2+16n}}{2n}&\text{ if }3\le n\le 6. \end{cases} \end{align}\] Theorem 1 in Section 2 is a result concerning global existence. Theorem 3 in Section 3 states that for any \(\gamma>0\) solutions blow-up in finite time if \(p\) is below the Fujita exponent. These two results is illustrated in Figure 1. We remark that the critical exponent coincides with the Fujita exponent in particular for \(n=1,2\).
Notation:
We list the notations used throughout this paper. Unless otherwise stated, the following conventions apply.
Constants denote positive real numbers and may change from line to line; \(C\) and \(C'\) denote a positive constants whose value may vary with context.
The notation \(X\lesssim Y\) means there exists \(C>0\) such that \(X\le C\,Y\). The notation \(X\simeq Y\) means \(X\lesssim Y\) and \(Y\lesssim X\).
For \(f\in\mathcal{S}'(\mathbb{R}^n)\), \(\mathcal{F}f\) or \(\hat{f}\) denotes its Fourier transform, and \(\mathcal{F}^{-1}f\) or \(\check f\) denotes the inverse transform.
For \(r>0\), set \(B(r):=\{x\in\mathbb{R}^n:\,|x|<r\}\). The symbol \(\chi_{|\cdot|<r}\) denotes the characteristic function of \(B(r)\).
For \(p>1\), the conjugate exponent \(p'>1\) is defined by \(\dfrac{1}{p}+\dfrac{1}{p'}=1\).
For \(1\le p\le\infty\), the Lebesgue space \(L^p(\mathbb{R}^n)\) is defined as usual:
\[L^p(\mathbb{R}^n):=\{f \mid \|f\|_{L^p}<\infty\},\]
where for \(1\le p<\infty\)
\[\|f\|_{L^p}:=\Big(\int_{\mathbb{R}^n}|f(x)|^p\,dx\Big)^{1/p},\]
and for \(p=\infty\)
\[\|f\|_{L^\infty}:=\operatorname{ess\,sup}_{x\in\mathbb{R}^n}|f(x)|.\]
For \(s\in\mathbb{R}\), the (inhomogeneous) Sobolev space \(H^s(\mathbb{R}^n)\) is defined by
\[H^s(\mathbb{R}^n):=\Big\{f\in\mathcal{S}'(\mathbb{R}^n)\;\Big|\;\|f\|_{H^s}:=\Big(\int_{\mathbb{R}^n}(1+|\xi|^2)^s|\widehat{f}(\xi)|^2\,d\xi\Big)^{1/2}<\infty\Big\},\]
and \(H^s(\mathbb{R}^n)\) is a Hilbert space with inner product
\[\langle f,g\rangle_{H^s}:=\int_{\mathbb{R}^n}(1+|\xi|^2)^s\widehat{f}(\xi)\overline{\widehat{g}(\xi)}\,d\xi.\]
Unless otherwise stated, all function spaces in this paper are taken on the whole space \(\mathbb{R}^n\); accordingly \(L^p\) and \(H^s\) denote \(L^p(\mathbb{R}^n)\) and \(H^s(\mathbb{R}^n)\) respectively.
2 Main results of global existence↩︎
This section presents a theorem obtained regarding the existence of global solutions.
We say that a function \(u\) is a mild solution of 1 if \(u\) satisfies \[\begin{align} u=K'(t)*u_0+K(t)*(u_1+u_0)+\int_0^tK(t-\tau)*|u|^p(\tau) d\tau, \end{align}\] where the function \(K\) is the fundamental solution of linear dumped wave equation written as \[\begin{align} \hat{K}=e^{-\frac{t}{2}}\sum_{k=0}^\infty \frac{\left(\frac{1}{4}-|\xi|^2\right)^k}{(2k+1)!}t^{2k+1}= \begin{cases} e^{-\frac{t}{2}}\frac{\sinh\left(t\sqrt{\frac{1}{4}-|\xi|^2}\right)}{\sqrt{\frac{1}{4}-|\xi|^2}}&|\xi|<\frac{1}{2}\\ e^{-\frac{t}{2}}&|\xi|=\frac{1}{2}\\ e^{-\frac{t}{2}}\frac{\sin\left(t\sqrt{|\xi|^2-\frac{1}{4}}\right)}{\sqrt{|\xi|^2-\frac{1}{4}}}&|\xi|>\frac{1}{2} \end{cases}. \end{align}\]
Theorem 1. Let \(1\le n\le6\) and let \(s\in(0,1]\) suppose that \[\begin{align} \gamma\ge \min\left\{\frac{n}{2}, \frac{\sqrt{n^2+16n}-n}{4}\right\}= \begin{cases} \frac{n}{2}&n=1\text{ or }2\\ \frac{\sqrt{n^2+16n}-n}{4}&n\ge3 \end{cases},\label{ga} \end{align}\qquad{(1)}\] \[\begin{align} p>\max\left\{1+\frac{2}{n},\frac{\sqrt{n^2+16n}+n}{2n}\right\}= \begin{cases} 1+\frac{n}{2} & n=1\text{ or }2\\ \frac{\sqrt{n^2+16n}+n}{2n}& n\ge3 \end{cases},\label{pga} \end{align}\qquad{(2)}\] and \(p\le\frac{n}{n-2s}\) if \(n>2s\). If \[\begin{align} (u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2), \end{align}\]then, for a sufficiently small \(\epsilon\), there is a unique mild solution \[\begin{align} u\in C([0,\infty):H^{s}) \end{align}\] to the Cauchy problem for the semilinear damped wave equation 1 . Furthermore, for the number \(\tilde{\gamma}>0\) satisfing \[\begin{align} \tilde{\gamma}<\frac{n}{2}\label{ga391} \end{align}\qquad{(3)}\] and \[\begin{align} \tilde{\gamma}\le\min\left\{\frac{n(p-1)}{2}, \gamma\right\},\label{ga392} \end{align}\qquad{(4)}\] the solution \(u\) satisfy the following decay estimates: \[\begin{align} \|u(t,\cdot)\|_{L^2}&\lesssim \epsilon(1+t)^{-\frac{\tilde{\gamma}}{2}}\left(\|(u_0,u_1)\|_{({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)}\right)\\ \|u(t,\cdot)\|_{H^s}&\lesssim \epsilon(1+t)^{-\frac{s+\tilde{\gamma}}{2}}\left(\|(u_0,u_1)\|_{({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)}\right). \end{align}\]
The proof of this theorem utilizes the global existence result established in [1], together with the fact that, for \(0<\gamma<\tilde{\gamma}\) and \(s\ge0\), one has \(\dot{H}^{-\tilde{\gamma}}\cap H^s\subset {\dot{H}^{-\gamma}}\cap H^s\) . The specific method will be described in Section 4.
3 Main results of blow-up solution↩︎
In this section, we establish three blow-up theorems. They are distinguished by the assumptions imposed on \(p\) and the initial data, and they provide different lifespan estimates as a result.
Theorem 2. Let \(n\in\mathbb{N}, \gamma>0\) and \(1<p<1+\frac{4}{n+2\gamma}\). Assume that the initial data \((u_0,u_1)\in(H^s\cap {\dot{H}^{-\gamma}})\times(L^2\cap{\dot{H}^{-\gamma}})\) satisfy \(\widehat{u_0+u_1}\ge0\) and \[\begin{align} \widehat{u_0+u_1}(\xi)\gtrsim|\xi|^{-\frac{n}{2}+\gamma}(\log(|\xi|))^{-1}\chi_{|\cdot|<r}(\xi),\label{inHg} \end{align}\qquad{(5)}\] for some \(r>0\). Then, the Cauchy problem 1 has no global (in time) weak solution. Moreover, the lifespan of the local solution \(T\) satisfies \[\begin{align} T\lesssim\epsilon^{-\frac{1}{p'-1-\frac{\gamma}{2}-\frac{n}{4}-\delta}}\label{lifespan1} \end{align}\qquad{(6)}\] for any \(0<\delta\ll1\).
Remark 1. If the initial data satisfy \(\widehat{u_0}=\widehat{u_1}=|\xi|^{-\frac{n}{2}+\gamma}(\log(|\xi|))^{-1}\chi_{|\cdot|<r}(\xi)\), then \((u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\).
Remark 2. The value \(-1/(p'-\gamma/2-n/4)\) that appears in the lifespan estimate ?? can also be written as\(-1/(p'-p'_{crit})\).
Theorem 3. Let \(n\in\mathbb{N}, 1<p<1+\frac{2}{n}\) and \(0<\epsilon\ll1\). Assume that the initial data \((u_0,u_1)\in(H^s\cap {\dot{H}^{-\gamma}})\times(L^2\cap{\dot{H}^{-\gamma}})\) satisfy \(\widehat{u_0+u_1}\ge0\) and \[\begin{align} \widehat{u_0+u_1}(\xi)>0\;\text{a.e. }\xi\in B(r), \end{align}\] for some \(r>0\). Here \(B(r):=\{\xi\in\mathbb{R}^n: |\xi|<r\}\). Then, the Cauchy problem 1 has no global (in time) weak solution. Moreover, the lifespan of the local solution \(T\) satisfies \[\begin{align} T\lesssim\epsilon^{-\frac{p}{p'-1-\frac{n}{2}}}. \end{align}\]
Remark 3. If the initial data satisfy \(u_0=u_1=\Delta^k e^{-\frac{|x|^2}{2}}\) with \(k\ge\gamma/2\), then \((u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) and \((u_0,u_1)\) satisfy the conditions of Theorem 3.
Remark 4. The value \(-p/(p'-n/2)\) that appears in the lifespan estimate ?? can also be written as\(-p/(p'-p'_\mathrm{F})\).
Theorem 4. Let \(n\in\mathbb{N}, 1<p\) and \(0<\epsilon\ll1\). Assume that the initial data \((u_0,u_1)\in(H^s\cap {\dot{H}^{-\gamma}})\times(L^2\cap{\dot{H}^{-\gamma}})\) satisfy \(\widehat{u_0+u_1}\ge0\), \[\begin{align} 1+\frac{n}{4}p<p' \end{align}\]i.e. \[\begin{align} p>\frac{n+\sqrt{n^2+16n}}{2n} \end{align}\] and \[\begin{align} \widehat{u_0+u_1}(\xi+a)\gtrsim|\xi|^{-n/2}(\log(|\xi|^{-1}))\;\text{a.e. }\xi\in B(r)\label{inHg2} \end{align}\qquad{(7)}\] for some \(r>0\) and \(0\neq a\in\mathbb{R}^n\). Then, the Cauchy problem 1 has no global (in time) weak solution. Moreover, the lifespan of the local solution \(T\) satisfies \[\begin{align} T\lesssim\epsilon^{-\frac{p}{p'-1-\frac{n}{4}p-\delta}}\label{b-u3l} \end{align}\qquad{(8)}\] for \(0<\delta\ll1\).
Remark 5. By Theorem 2, Theorem 3 and Theorem 4, we get the upper estimate of lifespan as \[\begin{align} T\lesssim\min\left\{\epsilon^{-\frac{p}{p'-p'_{\mathrm{F}}}}, \epsilon^{-\frac{1}{p'-p'_{\mathrm{crit}}-\delta}}, \epsilon^{-\frac{p}{p'-1-\frac{n}{4}p-\delta}} \right\}= \begin{cases} \epsilon^{-\frac{p}{p'-p'_{\mathrm{F}}}} &p\ge \max \{2,\frac{4+2n}{n+2\gamma}\}\\ \epsilon^{-\frac{1}{p'-p'_{\mathrm{crit}}-\delta}} & p\le \min\{\frac{4+2n}{n+2\gamma}, \frac{2}{\gamma}\}\\ \epsilon^{-\frac{p}{p'-1-\frac{n}{4}p-\delta}} & \frac{2}{\gamma}<p\le2 \end{cases} \end{align}\] for \(1<p<p_{\mathrm{crit}}\), which is illustrated in Figure 2.
4 Proof of global existence↩︎
In this section, we give the proof of Theorem 1. Before proceeding, we state the following result of [1].
Lemma 1 ([1]). Let \(n\le6\), \(s\in(0,1]\) and \(0<\gamma<n/2\). Let \(p\) fulfil \(p>1+\frac{4}{2\gamma+n}\), \(p\ge 1+\frac{2\gamma}{n}\) and \(p\le\frac{n}{n-2s}\) if \(n>2s\). Let us assume \[\begin{align} (u_0,u_1)\in({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2). \end{align}\]Then, for a sufficiently small \(\epsilon\), there is a unique mild solution \[\begin{align} u\in C([0,\infty):H^{s}) \end{align}\] to the Cauchy problem for the semilinear damped wave equation 1 . Furthermore, the solution \(u\) satisfies the following decay estimates: \[\begin{align} \|u(t,\cdot)\|_{L^2}&\lesssim \epsilon(1+t)^{-\frac{\gamma}{2}}\left(\|(u_0,u_1)\|_{({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)}\right)\\ \|u(t,\cdot)\|_{H^s}&\lesssim \epsilon(1+t)^{-\frac{s+\gamma}{2}}\left(\|(u_0,u_1)\|_{({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)}\right). \end{align}\]
Before proving Theorem 1, we state that \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) is monotonic with the parametar \(\gamma\).
Lemma 2. Let \(s\ge0\) be fixed. Then for any parameters satisfying \(0\le\tilde{\gamma}<{\gamma}\), the following inclusion holds: \[\begin{align} H^s\cap\dot{H}^{-{\gamma}}\subset H^s\cap\dot{H}^{-\tilde{\gamma}}. \end{align}\]
Proof of Lemma 2. \[\begin{align} &\|f\|_{H^{s}\cap \dot{H}^{-\tilde{\gamma}}}\\ \simeq&\|\langle\xi\rangle^{s}\hat{f}\|_{L^2}+\||\xi|^{-\tilde{\gamma}}\hat{f}\|_{L^2}\\ \lesssim&\|\chi_{|\xi|\ge1}(\xi)\langle\xi\rangle^{s}\hat{f}\|_{L^2}+\|\chi_{|\cdot|\le1}(\xi)|\xi|^{-\tilde{\gamma}}\|_{L^2}\\ \le&\|\chi_{|\cdot|\ge1}(\xi)\langle\xi\rangle^{s}\hat{f}\|_{L^2}+\|\chi_{|\cdot|\le1}(\xi)|\xi|^{-\gamma}\|_{L^2}\\ \lesssim&\|f\|_{H^{s}\cap \dot{H}^{-\gamma}}. \end{align}\] ◻
Proof of Theorem 1. We may assume without loss of generality that \[\begin{align} 2p'-2-\frac{n}{2}<\tilde{\gamma}. \end{align}\]
By ?? and Lemma 2, we get \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\subset(H^s\cap\dot{H}^{-\tilde{\gamma}})\times(L^2\cap\dot{H}^{-\tilde{\gamma}})\). By ?? , ?? and ?? , \(\tilde{\gamma}\) satisfies the conditions of the parameter \(\gamma\) in Lemma 1. Hence, we invoke Lemma 1 with \(\gamma\) repleced \(\gamma'\) in ?? and ?? to get Theorem 1. ◻
5 Proof of blow-up theorems↩︎
In this section, we give the proof of Theorem 2 and Theorem 3.
5.1 Construction of a bump function used in the proof↩︎
We need a non-trivial bump function \(\phi\in C_0^{\infty}(\mathbb{R}^n)\) which satisfies the following conditions. For the proof, we need some bump function given by the following lemma.
Lemma 3. There exists a non-trivial function \(\phi\in C_0^{\infty}(\mathbb{R}^n)\) which satisfies the following conditions:
\(\phi\ge0\) .
\(\widehat{\phi}\ge0\) .
\(\phi(Rx)\le\phi(rx)\) for all \(0<r<R\text{ and } x\in\mathbb{R}^n\).
Proof. Let \(\tilde{\phi}\in C_0^\infty(\mathbb{R}^n)\) satisfy [i], [iii] and \[\begin{align} |x|=|y|\Rightarrow\tilde{\phi}(x)=\tilde{\phi}(y)\;\;\;\text{for all } x,y\in\mathbb{R}^n.\label{newcon} \end{align}\tag{4}\] Let \(\eta\in C_0^{\infty}\left([0,\infty)\right)\) satisfy \(\tilde{\phi}(x)=\eta(|x|)\). Then, \(\eta'\le0\). For example, function \(f\) such that \[\begin{align} f(x)= \begin{cases} e^{-\frac{1}{1-|x|^2}}&|x|<1\\ 0&|x|\ge1 \end{cases}. \end{align}\]Then, \(\tilde{\phi}\) is a real-valued function. Now define the bump function \(\phi\mathrel{\vcenter{:}}=\tilde{\phi}*\tilde{\phi}\). We will show that \(\phi\) satisfies the conditions [i], [ii] and [iii].
Because of \(\tilde{\phi}\ge0\), we have \(\phi=\tilde{\phi}*\tilde{\phi}\ge0\).
\(\widehat{\phi}\) satisfies that \(\widehat{\phi}=\widehat{\tilde{\phi}*\tilde{\phi}}=\widehat{\tilde{\phi}}^2\). Where, \(\tilde{\phi}\) is real-valued and an even function because of 4 . Therefore, \(\widehat{\tilde{\phi}}\) is real-valued and \(\widehat{\phi}\) is non-negative function.
To prove (iii), it is sufficient to prove that \(\partial_R(\phi(Rx))\le0\). \[\begin{align} &\partial_R(\phi(Rx))\\ =&x\cdot(\nabla\phi)(Rx)\\ =&\int_{\mathbb{R}^n}x\cdot(\nabla\tilde{\phi})(y)\tilde{\phi}(Rx-y)dy\\ =&\int_{\mathbb{R}^n}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)\tilde{\phi}(Rx-y)dy\\ =&\int_{x\cdot y\le0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)\tilde{\phi}(Rx-y)dy +\int_{x\cdot y\ge0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)\tilde{\phi}(Rx-y)dy.\label{half} \end{align}\tag{5}\] To change the variable to \(y'=-y\), the first half of 5 is \[\begin{align} \int_{x\cdot y'\ge0}x\cdot\left(\frac{-y'}{|y'|}\eta'(|y'|)\right)\tilde{\phi}(Rx+y')dy'. \end{align}\] So, with attention for \(|y|=|-y|\), 5 is \[\begin{align} &-\int_{x\cdot y\ge0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)\tilde{\phi}(Rx+y)dy +\int_{x\cdot y\ge0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)\tilde{\phi}(Rx-y)dy\\ =&\int_{x\cdot y\ge0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)(\tilde{\phi}(Rx+y)-\tilde{\phi}(Rx+y))dy\\ =&\int_{x\cdot y\ge0}x\cdot\left(\frac{y}{|y|}\eta'(|y|)\right)(\eta(|Rx-y|)-\eta(|Rx+y|))dy\\ =&\int_{x\cdot y\ge0}\left(x\cdot\frac{y}{|y|}\right)\eta'(|y|)(\eta(|Rx-y|)-\eta(|Rx+y|))dy.\label{le0} \end{align}\tag{6}\] On the integrand function in 6 , since \(x\cdot y\ge0\). we have \(|Rx-y|\le|Rx+y|\). Thus, \(\eta(|Rx-y|)-\eta(|Rx+y|)\ge0\). Moreover, we have \(x\cdot(y/|y|)\ge0\) and \(\eta'(|y|)\le0\). So, we obtain \(\eqref{le0}\le0\).
◻
Lemma 4. Let \(\phi\in C_0^{\infty}(\mathbb{R}^n)\) satisfy the conditions [i], [ii] and [iii]. Then, for all \(k\in\mathbb{N}\), \(\phi^k\) also satisfies those conditions.
Proof. We will show that \(\phi^k\) satisfies [i], [ii] and [iii]. Then, since
Since \(\phi\ge0\), we get \(\phi^k\ge0\).
We have \[\begin{align} \widehat{\phi^k}=\widehat{\phi}*\ldots*\widehat{\phi}. \end{align}\]Then, since \(\widehat{\phi}\ge0\), We get \(\widehat{\phi^k}\ge0\).
Since \(\phi\) satisfies [iii], we get that \(\phi^k\) satisfies [iii].
◻
5.2 Proof of the blow-up theorems↩︎
Let \(u\) be the local weak solution of 1 . If the lifespan \(T\) is in \([0,1]\), then we get \(T\le\epsilon^{-a}\text{ for all }a>0\text{ and }0<\epsilon\ll1\). Therefore, let us assume \(T>1\). Let \(\tilde{\eta}\in C_0^\infty{[0,\infty)}\) be a monotonically non-increasing function that satisfies \(\mathop{\mathrm{supp}}\tilde{\eta}\subset[0,1]\) and \(\tilde{\eta}(t)=1 \text{ for all }t\in[0,1/2]\). Let \(\tilde{\phi}\in C_0^{\infty}(\mathbb{R}^n)\) satisfy [i], [ii] and [iii]. We take \(l\in\mathbb{N}\) being \(l>2p'\) and we set \(\phi\mathrel{\vcenter{:}}=\tilde{\phi}^l\) and \(\eta\mathrel{\vcenter{:}}=\tilde{\eta}^l\). We set \({\phi}_R(x)\mathrel{\vcenter{:}}=\tilde{\phi}(R^{-1}x)\text{ and }\eta_R(t)\mathrel{\vcenter{:}}=\eta(R^{-2}t)\) for all \(R>0\). Let a function \(I:[1,\sqrt{T})\to\mathbb{R}\) be \[\begin{align} I(R)\mathrel{\vcenter{:}}=\iint_{\mathbb{R}^n\times[0,T)}|u(x,t)|^p\phi_R(x)\eta_R(t)dxdt.\label{I} \end{align}\tag{7}\]
First, we will show a lemma for the theorems of blow-up.
Lemma 5. The function \(I(R)\) satisfies \[\begin{align} I(R)\le-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_R(x)dx+CI(R)^{1/p}R^{\frac{n+2}{p'}-2}\label{lem} \end{align}\qquad{(9)}\] and \[\begin{align} I(R)\lesssim-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_R(x)dx+CR^{n+2-2p'}\label{lem2} \end{align}\qquad{(10)}\] for all \(R\in[1,\sqrt{T})\) with a constantant \(C>0\) .
Proof of Lemma 5. First, since \(u\) is weak solution of the Cauchy problem 1 , we have \[\begin{align} &I(R)\\ =&-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_R(x)dx\\ &+\iint_{\mathbb{R}^n\times[0,T)}u\left(\partial^2_t-\partial_t-\Delta\right)(\phi_R\eta_R)dxdt\label{27}. \end{align}\tag{8}\] We further estimate the term 8 as \[\begin{align} &\iint_{\mathbb{R}^n\times[0,T)}u\left(\partial^2_t-\partial_t-\Delta\right)(\phi_R\eta_R)dxdt\\ =&\iint_{\mathop{\mathrm{supp}}\phi_R\eta_R}\left(|u|^p\phi_R\eta_R\right)^{1/p}\phi_R^{-1/p}\eta_{R}^{-1/p}\left(\partial^2_t-\partial_t-\Delta\right)(\phi_R\eta_R)dxdt\\ \le&I(R)^{1/p}\left(\iint_{\mathop{\mathrm{supp}}\phi_R\eta_R}\phi_R^{-p'/p}\eta_R^{-p'/p}\left|\left(\partial^2_t-\partial_t-\Delta\right)(\phi_R\eta_R)\right|^{p'}dxdt\right)^{1/p'}\;\;(\text{by H\"older's inequality}).\\ =&I(R)^{1/p}\left(\iint_{\mathop{\mathrm{supp}}\phi_R\eta_R}\phi^{-p'/p}\eta^{-p'/p}\left|\left(R^{-4}\frac{\partial^2}{\partial\tau^2}-R^{-2}\frac{\partial}{\partial\tau}-R^{-2}\Delta\right)(\phi\eta)\right|^{p'}R^{n+2}dyd\tau\right)^{1/p'}\\ \lesssim&I(R)^{1/p}R^{\frac{n+2}{p'}-2}\left(\iint_{\mathop{\mathrm{supp}}\phi\eta}\phi^{-p'/p}\eta^{-p'/p}\left|\left(\frac{\partial^2}{\partial\tau^2}-\frac{\partial}{\partial\tau}-\Delta\right)(\phi\eta)\right|^{p'}dyd\tau\right)^{1/p'}\label{Integral32Check} \end{align}\tag{9}\] Here, we will show the value \[\begin{align} \iint_{\mathop{\mathrm{supp}}\phi\eta}\phi^{-p'/p}\eta^{-p'/p}\left|\left(\frac{\partial^2}{\partial\tau^2}-\frac{\partial}{\partial\tau}-\Delta\right)(\phi\eta)\right|^{p'}dyd\tau \end{align}\]is finite. The integrand function is \[\begin{align} &\phi^{-p'/p}\eta^{-p'/p}\left|\left(\frac{\partial^2}{\partial\tau^2}-\frac{\partial}{\partial\tau}-\Delta\right)(\phi\eta)\right|^{p'}\\ =&\left|\phi^{-1/p}\eta^{-1/p}\left(\frac{\partial^2}{\partial\tau^2}-\frac{\partial}{\partial\tau}-\Delta\right)(\phi\eta)\right|^{p'}\label{24} \end{align}\tag{10}\] and, since the integral domain of 9 is bounded, it is sufficient to show that 10 is bounded. To do this, consider \(\phi=\tilde{\phi}^l,\;\eta=\tilde{\eta}^l\), and express 10 in terms of \(\tilde{\phi},\tilde{\eta}\) instead of \(\phi,\eta\). We will express \(\eta',\eta'',\Delta\phi\) as \(\tilde{\eta},\tilde{\phi}\). \[\begin{align} &\eta'=(\tilde{\eta}^l)'=l\tilde{\eta}'\tilde{\eta}^{l-1}\\ &\eta''=(\tilde{\eta}^l)''=l(l-1)(\tilde{\eta}')^{2}\tilde{\eta}^{l-2}+l\tilde{\eta}''\tilde{\eta}^{l-1}\\ &\Delta\phi=\Delta(\tilde{\phi}^l)=l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l-1}. \end{align}\]And, we get \[\begin{align} &\phi^{-1/p}\eta^{-1/p}\frac{\partial^2}{\partial\tau^2}(\phi\eta)\\ =&\tilde{\phi}^{-l/p}\tilde{\eta}^{-l/p}\tilde{\phi}^l(l(l-1)(\tilde{\eta}')^{2}\tilde{\eta}^{l-2}+l\tilde{\eta}''\tilde{\eta}^{l-1})\\ =&\tilde{\phi}^{l(1-1/p)}(l(l-1)(\tilde{\eta})')^2\tilde{\eta}^{l(1-1/p)-2}+l\tilde{\eta}''\tilde{\eta}^{l(1-1/p)-1})\\ =&\tilde{\phi}^{l/p'}(l(l-1)(\tilde{\eta})')^2\tilde{\eta}^{l/p'-2}+l\tilde{\eta}''\tilde{\eta}^{l/p'-1}),\tag{11}\\ &\phi^{-1/p}\eta^{-1/p}\frac{\partial}{\partial\tau}(\phi\eta)\\ =&\tilde{\phi}^{-l/p}\tilde{\eta}^{-1/p}\tilde{\phi}^{l}(l\tilde{\eta}'\tilde{\eta}^{l-1})\\ =&l\tilde{\phi}^{l/p'}\tilde{\eta}^{l/p'-1}\tilde{\eta}'\tag{12},\\ &\phi^{-1/p}\eta^{-1/p}\Delta(\phi\eta)\\ =&\tilde{\phi}^{-l/p}\tilde{\eta}^{-1/p}(l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l-1})\tilde{\eta}^{l}\\ =&\tilde{\eta}^{l(1-1/p)}(l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l(1-1/p)-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l(1-1/p)-1})\\ =&\tilde{\eta}^{l/p'}(l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l/p'-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l/p'-1})\tag{13} \end{align}\] \(\eqref{24}=|\eqref{29}+\eqref{32}+\eqref{36}|^{p'}\) and, the values appearing in the powers of 11 , 12 , and 13 are all positive (note that \(l>2p'\)). So 10 is bounded, and we get ?? .
Next, we will show ?? to use ?? . Applying Young’s inequality, the second term in ?? is bounded by \(CR^{n+2-p'}+I(R)/p\). ◻
Proof of Theorem 2. First, we will show that there exists a function in \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) which satisfies the condition ?? . It is sufficient to show the next lemma.
Lemma 6. With the conditions of Theorem 2, we have \[\begin{align} \mathcal{F}^{-1}\left(|\xi|^{-\frac{n}{2}+\gamma}(\log(|\xi|^{-1}))^{-1}\chi_{|\cdot|<r}(\xi)\right)\in L^2\times{\dot{H}^{-\gamma}}. \end{align}\]
Proof of Lemma 6. It is sufficient to show that \[\begin{align} f(\xi):=|\xi|^{-\frac{n}{2}+\gamma}(\log(|\xi|^{-1}))^{-1}\chi_{|\cdot|<r}(\xi)\in L^2 \end{align}\] and \[\begin{align} g(\xi):=|\xi|^{-\frac{n}{2}}(\log(|\xi|^{-1}))^{-1}\chi_{|\cdot|<r}(\xi)\in L^2.\label{f40xi41} \end{align}\tag{14}\] To show them, it is sufficient to show only \(g\in L^2\). We have \[\begin{align} &\int_{\mathbb{R}^n}|g(\xi)|^2d\xi\\ =&\int_{\mathbb{R}^n}|\xi|^{-n}(\log(|\xi|))^{-2}\chi_{|\cdot|<1}(\xi)d\xi\\ =&\int_{|\xi|<r}|\xi|^{-n}(\log(|\xi|))^{-2}d\xi.\label{xi61romega} \end{align}\tag{15}\] by change of variable \(\xi=e^{l}\omega(-\infty<l<0,\omega/r\in S^{n-1})\), we get \[\begin{align} \eqref{xi61romega}&=\iint_{l<0,\omega/r\in S^{n-1}}e^{-nl}l^{-2}e^{nl}dld\omega\\ &\simeq\int_{-\infty}^0l^{-2}dl<\infty. \end{align}\] ◻
With the conditions of initial data, \[\begin{align} &\int_{\mathbb{R}^n}(u_0+u_1)\phi_R(x)dx\\ =&\int_{\mathbb{R}^n}\widehat{u_0+u_1}\check{\phi_R}(\xi)d\xi\\ =&\int_{\mathbb{R}^n}\widehat{u_0+u_1}(\xi)R^{n}\check{\phi}(R\xi)d\xi\\ =&\int_{\mathbb{R}^n}\widehat{u_0+u_1}(R^{-1}\xi)\check{\phi}(\xi)d\xi\\ \gtrsim&\int_{|\xi|\le rR}R^{\frac{n}{2}-\gamma}|\xi|^{-\frac{n}{2}+\gamma}(\log({R|\xi|^{-1}}))^{-1}\check{\phi}(\xi)d\xi.\label{ini2} \end{align}\tag{16}\] Here, the term \((\log({R|\xi|^{-1}}))^{-1}\) satisfies (note \(|\xi|\le rR\)) \[\begin{align} (\log({R|\xi|^{-1}}))^{-1}&=\left(\log R+\log(|\xi|^{-1})\right)^{-1}\\ &=\left(\log R\left(1+\frac{\log(|\xi|^{-1})}{\log R}\right)\right)^{-1}\\ &\ge \left(\log R\left(1+\frac{\log(|\xi|^{-1})}{\log R}\right)\right)^{-1}\\ &\ge \left(\log R\left(1+\frac{\log(|\xi|^{-1})}{\log( r^{-1}|\xi|)}\right)\right)^{-1}\\ &= \left(\log R\right)^{-1}\left(\frac{\log( r^{-1}|\xi|)+\log(|\xi|^{-1})}{\log( r^{-1}|\xi|)}\right)^{-1}\\ &= \left(\log R\right)^{-1}\left(\frac{\log( r^{-1})}{\log( r^{-1}|\xi|)}\right)^{-1}\\ &= \left(\log R\right)^{-1}\frac{\log( r^{-1}|\xi|)}{\log( r^{-1})}. \end{align}\] Therefore, we get \[\begin{align} &\int_{\mathbb{R}^n}(u_0+u_1)\phi_R(x)dx\gtrsim R^{\frac{n}{2}-\gamma}(\log R)^{-1} \end{align}\]and \[\begin{align} I(R)\lesssim-\epsilon R^{\frac{n}{2}-\gamma}(\log R)^{-1}+CR^{n+2-2p'}. \end{align}\]Since \(I(R)\ge0\), we get \(-\epsilon R^{\frac{n}{2}-\gamma}(\log R)^{-1}+CR^{n+2-2p'}\ge0\) and \[\begin{align} \epsilon\lesssim R^{\frac{n}{2}+2-2p'+\gamma}(\log R)\le R^{\frac{n}{2}+\gamma+2-2p'+\delta}\stackrel{R\to \sqrt{T}}{\longrightarrow}T^{\frac{n}{4}+\frac{\gamma}{2}+1-p'+\delta}. \end{align}\]So, we get \(T\lesssim\epsilon^{-\frac{1}{p'-(1+n/4+\gamma/2)}}\). ◻
Proof of Theorem 3. First we note \[\begin{align} &\int_{\mathbb{R}^n}(u_0+u_1)\phi_Rdx\\ =&\int_{\mathbb{R}^n}\widehat{u_0+u_1}\check{\phi_R}(\xi)d\xi.\label{ini620} \end{align}\tag{17}\] Since \(\check{\phi_R}(0)>0\) (because of [iii] and the fact that \(\phi_R\) is a positive non-trivial function with compact support ) and \(\hat{u_0}, \hat{u_1}>0\), we get \(\eqref{ini620}>0\). For ?? and \(I(R)>0\), we get \[\begin{align} 0&\lesssim-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_Rdx+CI(R)^{1/p}R^{\frac{n+2}{p'}-2} \end{align}\] and \[\begin{align} I(R)\ge\epsilon^{p}\left(\int_{\mathbb{R}^n}(u_0+u_1)\phi_RdxR^{\frac{n+2}{p'}-2}\right)^p(\eqqcolon \epsilon^pg(R)). \end{align}\] Because of the definition of \(I(R)\), \(I\) is a monotonically non-decreasing function. Therefore, we get \[\begin{align} \epsilon^pg(1)\lesssim I(1)\le I(R)&\lesssim-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_Rdx+CR^{n+2-2p'}\\ &\lesssim R^{n+2-2p'}\stackrel{R\to \sqrt{T}}{\longrightarrow} T^{\frac{n}{2}+1-p'}. \end{align}\]So, we get \(T\lesssim\epsilon^{-\frac{p}{p'-(1+n/2)}}\). ◻
Proof of Theorem 4. First, we will show that there exists a function in \(({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)\) which satisfies the condition ?? . It is sufficient to show the next lemma.
Lemma 7. With the conditions of Theorem 4, we have \[\begin{align} \mathcal{F}^{-1}\left(|\xi-a|^{-\frac{n}{2}}(\log(|\xi-a|))^{-1}\chi_{|\cdot|<1}(\xi-a)\right)\in L^2\times{\dot{H}^{-\gamma}}. \end{align}\]
Proof of Lemma 7. It is sufficient to show that \[\begin{align} \tilde{f}(\xi):=|\xi-a|^{-\frac{n}{2}}(\log(|\xi-a|))^{-1}\chi_{|\cdot|<r}(\xi-a)\in L^2\label{f40xi412} \end{align}\tag{18}\] and \[\begin{align} \tilde{g}(\xi):=|\xi|^{-\gamma}|\xi-a|^{-\frac{n}{2}}(\log(|\xi-a|))^{-1}\chi_{|\cdot|<r}(\xi-a)\in L^2.\label{g40xi412} \end{align}\tag{19}\] 18 is obvious form 14 by considering \(\tilde{f}(\xi+a)=g(\xi)\). Regarding 19 , Since \(0\notin\mathop{\mathrm{supp}}\tilde{g}\) (Take \(r\) in such a way that it is sufficiently small.), then \(|\xi^{-\gamma}|\) is bounded. Therefore \(\tilde{g}\in L^2\). ◻
First, we define the function \(J\) following \[\begin{align} J(R)\mathrel{\vcenter{:}}=\iint_{\mathbb{R}^n\times[0,T)}|u(x,t)|^p(2+\cos(a\cdot x))\phi_R(x)\eta_{1/2}(t)dxdt.\label{J} \end{align}\tag{20}\] The proof of this theorem uses the following lemma, which expresses the relationship between \(J\) and \(I\).
Lemma 8. The functions \(I(R),\;J(R)\) satisfy \[\begin{align} J(R)\lesssim-\epsilon R^{n/2}(\log R)^{-1}+CI(R)^{1/p}R^{n/p'}\label{lem3} \end{align}\qquad{(11)}\] for all \(R\in[1,\sqrt{T})\) with a constantant \(C>0\) .
Proof of Lemma 8. First, since \(u\) is weak solution of the Cauchy problem 1 , we have \[\begin{align} &J(R)\\ =&-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)(2+\cos(a\cdot x))\phi_R(x)dx\tag{21}\\ &+\iint_{\mathbb{R}^n\times[0,1)}u\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})dxdt\tag{22}. \end{align}\] We further estimate the term 8 as \[\begin{align} &\iint_{\mathbb{R}^n\times[0,T)}u\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})dxdt\\ =&\iint_{\mathop{\mathrm{supp}}\phi_R\eta_R}\left(|u|^p\phi_R\eta_R\right)^{1/p}\phi_R^{-1/p}\eta_{R}^{-1/p}\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})dxdt\\ \le&I(R)^{1/p}\left(\iint_{\mathop{\mathrm{supp}}\phi_R\eta_{1/2}}\phi_R^{-p'/p}\eta_R^{-p'/p}\left|\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})\right|^{p'}dxdt\right)^{1/p'}.\label{etain} \end{align}\tag{23}\] Considering the values of the functions \(\eta_{1/2}\) and \(\eta_R\), we can see that \(\eta_R=1\) with \(t\in\mathop{\mathrm{supp}}\eta_{1/2}\). So, 23 is \[\begin{align} &I(R)^{1/p}\left(\iint_{\mathop{\mathrm{supp}}\phi_R\eta_{1/2}}\phi_R^{-p'/p}\left|\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})\right|^{p'}dxdt\right)^{1/p'}.\label{etain2} \end{align}\tag{24}\] This leads us to examine \((2+\cos(a\cdot x))\phi_R\). Calculation yields \[\begin{align} &|\Delta((2+\cos(a\cdot x))\phi_R(x))|\\ =&|\Delta(2+\cos(a\cdot x))\phi_R(x)+2\nabla(2+\cos(a\cdot x))\cdot\nabla\phi_R(x)+(2+\cos(a\cdot x))\Delta\phi_R|\\ \le&|a|^2|\cos(a\cdot x)|\phi_R(x)+R^{-1}|\sin(a\cdot x)a||(\nabla\phi)(R^{-1}x)|+R^{-2}(2+\cos(a\cdot x))(\Delta\phi)(R^{-1}x)\\ \lesssim&\phi(R^{-1}x)+|(\nabla\phi)(R^{-1}x)|+|(\Delta\phi)(R^{-1}x)|. \end{align}\] There, we define \(g:=\phi+|\nabla\phi|+|\Delta\phi|\). And, we can get \((2+\cos(a\cdot x))\phi_R\lesssim\phi_R(x)\). Substituting these results into the integral in 24 , we obtain \[\begin{align} &\iint_{\mathop{\mathrm{supp}}\phi_R\eta_{1/2}}\phi_R^{-p'/p}\left|\left(\partial^2_t-\partial_t-\Delta\right)((2+\cos(a\cdot x))\phi_R\eta_{1/2})\right|^{p'}dxdt\\ \lesssim&\iint_{\mathop{\mathrm{supp}}\phi_R\eta_{1/2}}\phi_R|\eta''_{1/2}-\eta'_{1/2}|^{p'}dxdt\\ &+\iint_{\mathop{\mathrm{supp}}\phi_R\eta_{1/2}}\eta_{1/2}^{p'}\left(\phi_R+\phi^{-1/p}|(\nabla\phi)(R^{-1}x)|+\phi^{-1/p}|(\Delta\phi)(R^{-1}x)|\right)^{p'}dxdt\\ \lesssim&R^{n}\iint_{\mathop{\mathrm{supp}}\phi\eta_{1/2}}\phi|\eta''_{1/2}-\eta'_{1/2}|^{p'}dxdt\\ &+R^n\iint_{\mathop{\mathrm{supp}}\phi\eta_{1/2}}\eta_{1/2}^{p'}\left(\phi+\phi^{-1/p}|\nabla\phi|+\phi^{-1/p}|\Delta\phi|\right)^{p'}dxdt.\label{uxfheinc} \end{align}\tag{25}\]
Here we verify that the integral of 9 is finite. Since this integral domain is bounded, it is sufficient to show that 9 is bounded. To do this, consider \(\phi=\tilde{\phi}^l,\;\eta=\tilde{\eta}^l\), and express 9 in terms of \(\tilde{\phi}\) instead of \(\phi\). We will express \(\eta',\eta'',\Delta\phi\) as \(\tilde{\eta},\tilde{\phi}\). \[\begin{align} &\nabla\phi=\nabla(\tilde{\phi})=l\tilde{\phi}^{l-1}\nabla\tilde{\phi}\\ &\Delta\phi=\Delta(\tilde{\phi}^l)=l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l-1}. \end{align}\]And, we get \[\begin{align} \phi^{-1/p}|\nabla\phi|&=l\tilde{\phi}^{-l/p+l-1}|\nabla\phi|=l\tilde{\phi}^{l/p'-1}|\nabla\phi|\tag{26}\\ \phi^{-1/p}|\Delta\phi|&=l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{-l/p+l-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{-l/p+l-1}\\ &=l(l-1)|\nabla\tilde{\phi}|^2\tilde{\phi}^{l/p'-2}+l(\Delta\tilde{\phi})\tilde{\phi}^{l/p'-1}.\tag{27} \end{align}\] The values appearing in the powers of 11 and 12 are all positive (note that \(l>2p'\)). So 9 is bounded. So, we get \(\eqref{lemin}\lesssim I(R)^{1/p}R^{n/p'}\).
Next, we will check the estimate \(\eqref{term1}\lesssim-\epsilon R^{n/2}(\log R)^{-1}\). Considering Parseval’s identity, \[\begin{align} &\int_{\mathbb{R}^n}(u_0+u_1)(2+\cos(a\cdot x))\phi_R(x)dx\\ =&\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}(2+\cos(a\cdot x))\check{\phi_R}d\xi\\ =&\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}(\xi-a)\check{\phi_R}d\xi+2\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}(\xi)\check{\phi_R}d\xi+\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}(\xi+a)\check{\phi_R}d\xi\\ \ge&\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}(\xi+a)\check{\phi_R}d\xi\\ \gtrsim&\int_{|\xi|<r}|\xi|^{-2/n}(\log(|\xi|^{-1}))^{-1}\check{\phi_R}d\xi\\ =&\int_{|\xi|<rR}R^{n/2}|\xi|^{-n/2}(\log(R|\xi|^{-1}))^{-1}\check\phi(\xi)d\xi.\label{ini3} \end{align}\tag{28}\] The value 28 is equivalent to the value 16 with the case \(\gamma=0\). Therefore, by making the same argument as in Theorem2, we get \(\eqref{term1}\lesssim-\epsilon R^{n/2}(\log R)^{-1}\). ◻
By ?? and ?? , we can get \[\begin{align} 0\le J(R)\lesssim&-\epsilon R^{n/2}(\log R)^{-1}+CI(R)^{1/p}R^{n/p'}\\ \le&-\epsilon R^{n/2}(\log R)^{-1}+C\left(-\epsilon\int_{\mathbb{R}^n}(u_0+u_1)\phi_Rdx+CR^{n+2-2p'}\right)^{1/p}R^{n/p'}. \end{align}\] Here, Considering \[\begin{align} \int_{\mathbb{R}^n}(u_0+u_1)\phi_Rdx=\int_{\mathbb{R}^n}\widehat{(u_0+u_1)}\check\phi_Rdx>0\;\left(\text{by \eqref{ii} and \widehat{(u_0+u_1)}>0}\right), \end{align}\] we obtain \[\begin{align} 0&\le-\epsilon R^{n/2}(\log R)^{-1}+CR^{(n+2-2p')/p+n/p'}\\ \epsilon&\lesssim R^{(2-2p')/p+n/2}\log R\le R^{(2-2p')/p+n/2+\tilde{\delta}} \end{align}\] for any \(0<\tilde{\delta}\ll1\). By taking the limit \(R\to \sqrt{T}\) and appropriately selecting \(\tilde{\delta}\) in response to \(\delta\), we get the lifespan estimate ?? . ◻
Acknowledgment↩︎
I am deeply grateful to my supervisor, Prof. Mitsuru Sugimoto (Nagoya University), for his guidance throughout this work. I also thank Prof. Yuta Wakasugi (Hiroshima University) for helpful suggestions and for directing me to key references related to this work. I am grateful to Prof. Soichiro Suzuki (Chuo University) for providing the initial idea of condition ?? .
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Email: mitsuhiro.matsunaga.e6@math.nagoya-u.ac.jp