Finite time blow up for wave equations with strong damping in an exterior domain

TL;DR

This paper investigates conditions under which solutions to strongly damped wave equations in exterior domains become unbounded in finite time, focusing on the influence of initial data and the nonlinearity exponent p.

Contribution

It establishes blow-up results for strongly damped wave equations in exterior domains, providing new insights into the conditions leading to finite time blow-up.

Findings

Blow-up occurs under certain initial data conditions.

The exponent p influences the blow-up behavior.

Results extend understanding of damping effects in exterior domains.

Abstract

We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power type nonlinearity |u|^p. We will establish blow-up results under some conditions on the initial data and the exponent p.

Equations

$$\left\{\begin{array}[]{ll}\,\,\displaystyle{u_{tt}-\Delta u-\Delta u_{t}=|u|^{p}}&\displaystyle{t>0,x\in{\Omega},}\\ {}\\ \displaystyle{u(0,x)=u_{0}(x),\;\;u_{t}(0,x)=u_{1}(x)}&\displaystyle{x\in{\Omega},}\\ {}\\ \displaystyle{u=0,}&\displaystyle{t\geq 0,\;x\in{\partial\Omega},}\end{array}\right.$$

$$(u_0,u_1)\in (H^2(\Omega )\cap H_0^1(\Omega ))\times L^2(\Omega ),\text{mbox}and{u}_0,u_1\ge 0.$$

$$u\in C([0,T_{\max }),H_0^1(\Omega ))\cap C^1([0,T_{\max }),L^2(\Omega )),$$

$$\text{mbox}either{T}_{\max }=\infty \text{mbox}orelse{T}_{\max }<\infty \text{mbox}and\Vert u(t,\cdot ){\Vert }_{H^1}+\Vert u_t(t,\cdot ){\Vert }_2\to \infty \text{mbox}ast\to T_{\max }.$$

$${\int }_{\Omega }[u_1(x)-\Delta u_0(x)]{\varphi }_0(x)dx>0,$$

$$\left\{\begin{array}[]{l}\displaystyle 1<p\leq 3,\quad\mbox{for}\;n=1,\\ \displaystyle 1<p<3,\quad\mbox{for}\;n=2,\\ \displaystyle 1<p\leq 1+\frac{2}{n-1},\quad\mbox{for}\;n\geq 3,\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta\phi_{0}(x)=0,\;\mbox{in}\;\Omega,\;\;n\geq 3,\\ \phi_{0}|_{\partial\Omega}=0,\\ |x|\rightarrow\infty,\quad\phi_{0}(x)\rightarrow 1.\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta\phi_{0}(x)=0,\;\mbox{in}\;\Omega,\;\;n=2,\\ \phi_{0}|_{\partial\Omega}=0,\\ |x|\rightarrow\infty,\quad\phi_{0}(x)\rightarrow+\infty,\;\;\mbox{and}\;\phi_{0}(x)\;\mbox{increase at the rate of}\;\ln(|x|).\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta\phi_{0}(x)=0,\;x>0,\\ \phi_{0}|_{x=0}=0,\\ x\rightarrow\infty,\quad\phi_{0}(x)\rightarrow+\infty,\;\;\mbox{and}\;\phi_{0}(x)\;\mbox{increase at the rate of linear function $x$}.\end{array}\right.$$

$${\int }_0^T{\int }_{\Omega }|u{|}^p\phi dxdt+{\int }_{\Omega }[u_1(x)-\Delta u_0(x)]\phi (0,x)dx-{\int }_{\Omega }{u}_0(x){\phi }_t(0,x)dx$$

$$={\int }_0^T{\int }_{\Omega }u{\phi }_{tt}dxdt+{\int }_0^T{\int }_{\Omega }u\Delta {\phi }_{t}dxdt-{\int }_0^T{\int }_{\Omega }u\Delta \phi dxdt,$$

$$\eta(t):=\left\{\begin{array}[]{ll}\displaystyle{1}&\displaystyle{\quad\mbox{if }0\leq t\leq 1/4}\\ \displaystyle{0}&\displaystyle{\quad\mbox{if }t\geq 1,}\end{array}\right.$$

$$\Phi(r):=\left\{\begin{array}[]{ll}\displaystyle{1}&\displaystyle{\quad\mbox{if }0\leq r\leq 1}\\ \displaystyle{0}&\displaystyle{\quad\mbox{if }r\geq 2,}\end{array}\right.$$

$${\int }_0^T{\int }_{{\Omega }_1}|u{|}^p\phi dxdt+{\int }_{{\Omega }_1}[u_1(x)-\Delta u_0(x)]{\varphi }_0(x){\phi }_T^{\ell }(x)dx$$

$$={\int }_0^T{\int }_{{\Omega }_1}u{\varphi }_0(x){\phi }_T^{\ell }(x){\partial }_t^2({\eta }_T^k(t))dxdt+{\int }_0^T{\int }_{{\Omega }_1}u\Delta [{\varphi }_0(x){\phi }_T^{\ell }(x)]{\partial }_t({\eta }_T^k(t))dxdt-{\int }_0^T{\int }_{{\Omega }_1}u\Delta [{\varphi }_0(x){\phi }_T^{\ell }(x)]{\eta }_T^k(t)dxdt$$

$$=:I_1+I_2+I_3$$

$$I_1$$

$$I_2$$

$$I_3$$

$${\int }_{{\Omega }_1}[u_1(x)-\Delta u_0(x)]{\varphi }_0(x){\phi }_T^{\ell }(x)dx$$

$$\le \frac{1}{2}{\int }_0^T{\int }_{{\Omega }_1}|u{|}^p\phi dxdt+{\int }_{{\Omega }_1}[u_1(x)-\Delta u_0(x)]{\varphi }_0(x){\phi }_T^{\ell }(x)dx$$

$$\le C{\int }_0^T{\int }_{{\Omega }_1}{\varphi }_0(x){\phi }_T^{\ell }(x){\eta }_T(t{)}^{(k-2)p^{\prime }}{\left|{\partial }_t{\eta }_T(t)\right|}^{2p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{{\Omega }_1}{\varphi }_0(x){\phi }_T^{\ell }(x){\eta }_T(t{)}^{(k-1)p^{\prime }}{\left|{\partial }_t^2{\eta }_T(t)\right|}^{p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -p^{\prime }}(x){\eta }_T^{k-p^{\prime }}(t)|\nabla {\varphi }_0(x){|}^{p^{\prime }}{\left|\nabla {\phi }_T(x)\right|}^{p^{\prime }}|{\partial }_t({\eta }_T(t)){|}^{p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -2p^{\prime }}(x){\eta }_T^{k-p^{\prime }}(t){\left|\nabla {\phi }_T(x)\right|}^{2p^{\prime }}|{\partial }_t({\eta }_T(t)){|}^{p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -p^{\prime }}(x){\eta }_T^{k-p^{\prime }}(t){\left|\Delta {\phi }_T(x)\right|}^{p^{\prime }}|{\partial }_t({\eta }_T(t)){|}^{p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -p^{\prime }}(x){\eta }_T^k(t)|\nabla {\varphi }_0(x){|}^{p^{\prime }}{\left|\nabla {\phi }_T(x)\right|}^{p^{\prime }}dxdt$$

$$+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -2p^{\prime }}(x){\eta }_T^k(t){\left|\nabla {\phi }_T(x)\right|}^{2p^{\prime }}dxdt+C{\int }_0^T{\int }_{\nabla {\Omega }_1}{\phi }_T^{\ell -p^{\prime }}(x){\eta }_T^k(t){\left|\Delta {\phi }_T(x)\right|}^{p^{\prime }}dxdt,$$

$${\int }_{{\Omega }_1}[u_1(x)-\Delta u_0(x)]{\varphi }_0(x){\phi }_T^{\ell }(x)dx$$

$$0<{\int }_{\Omega }[u_1(x)-\Delta u_0(x)]{\varphi }_0(x)dx\le 0;$$

$${\int }_0^T{\int }_{{\Omega }_1}|u{|}^p\phi dxdt\le D,\text{mbox}forallT>0,$$

$${\int }_{T/2}^T{\int }_{{\Omega }_1}|u{|}^p\phi dxdt,{\int }_{T/2}^T{\int }_{\nabla {\Omega }_1}|u{|}^p\phi dxdt,{\int }_0^T{\int }_{\nabla {\Omega }_1}|u{|}^p\phi dxdt\to 0\text{mbox}asT\to \infty .$$

$$I_1$$

$$I_2$$

$$I_3$$

$${\int }_{{\Omega }_1}[u_1(x)-\Delta u_0(x)]{\varphi }_0(x){\phi }_T^{\ell }(x)dx$$

Full text

1 Introduction

This paper concerns the initial boundary value problem of the strongly damped wave equation in an exterior domain. Let $\Omega\subset\mathbb{R}^{n}$ be an exterior domain whose obstacle $\mathcal{O}\subset\mathbb{R}^{n}$ is bounded with smooth compact boundary $\partial\Omega$. We consider the initial boundary value problem

[TABLE]

where the unknown function $u$ is real-valued, $n\geq 1$, and $p>1.$ Throughout this paper, we assume that

[TABLE]

Without loss of generality, we assume that $0\in\mathcal{O}\subset\subset B(R)$, where $B(R):=\{x\in\mathbb{R}^{n}:\;|x|<R\}$ is a ball of radius $R$ centred at the origin.

For the simplicity of notations, $\|\cdotp\|_{q}$ and $\|\cdotp\|_{H^{1}}$ $(1\leq q\leq\infty)$ stand for the usual $L^{q}(\Omega)$-norm and $H^{1}_{0}(\Omega)$-norm, respectively.

First, the following local well-posedness result is needed.

Proposition 1**.**

[3, see Proposition 2.1]

Let $1<p<\infty$ for $n=1,2$ and $1<p\leq\frac{n}{n-2}$ for $n\geq 3$. Under the assumption $(\ref{condition1})$, there exists a maximal existence time $T_{\max}>0$ such that the problem $(\ref{eq1})$ possesses a unique weak solution

[TABLE]

where $0<T_{\max}\leq\infty.$ In addition:

[TABLE]

** Remark 1****.**

We say that $u$ is a global solution of $(\ref{eq1})$ if $T_{\max}=\infty,$ while in the case of $T_{\max}<\infty,$ we say that $u$ blows up in finite time.

Our main result is the following

Theorem 1**.**

Assume that the initial data satisfies $(\ref{condition1})$ such that

[TABLE]

where $\phi_{0}(x)$ is defined below (see Lemma 1, 2, 3). If

[TABLE]

then the solution of the problem $(\ref{eq1})$ blows up in finite time.

This paper is organized as follows: in Section 2, we present several preliminaries. Section 3 contains the proofs of the blow-up theorem (Theorem 1).

2 Preliminaries

In this section, we give some preliminary properties that will be used in the proof of Theorem 1.

Lemma 1**.**

There exists a function $\phi_{0}(x)\in C^{2}(\Omega)\cap C(\overline{\Omega})$ satisfying the following boundary value problem

[TABLE]

Moreover, $\phi_{0}(x)$ satisfies:

$0<\phi_{0}(x)<1$, for all $x\in\Omega$.

• 2.

$\phi_{0}(x)\geq C$, for all $|x|\gg 1$.

• 3.

$|\nabla\phi_{0}(x)|\leq\frac{C}{|x|^{n-1}}$, for all $|x|\gg 1$.

Proof.

From [4, Lemma 2.2] there exists a regular solution $\phi_{0}$ of (2.1) such that $0<\phi_{0}(x)<1$, for all $x\in\Omega$. To obtain the last two properties of $\phi_{0}$, it is easy to see that since $\mathcal{O}$ is bounded, there exist $r_{2}>r_{1}>0$ such that $B_{r_{1}}\subseteq\mathcal{O}\subseteq B_{r_{2}}$, where $B_{r}$ stands for the open ball with center zero and radius $r$. By the maximum principle we conclude that $\phi_{1}(x)\leq\phi_{0}(x)\leq\phi_{2}(x)$ in $\Omega$, where $\phi_{1}(x)$ and $\phi_{2}(x)$ are, respectively, the solution of (2.1) on $\mathbb{R}^{n}\setminus B_{r_{1}}$ and $\mathbb{R}^{n}\setminus B_{r_{2}}$. We remember tha $\phi_{i}(x)=r_{i}^{2-n}-|x|^{2-n}$, $i=1,2$. Moreover, the standard elliptic theory implies that $|\nabla\phi_{0}(x)|\sim|\nabla\phi_{i}(x)|$, $i=1,2$. As $\phi_{1}(x)|\geq C$ and $|\nabla\phi_{i}(x)|\leq\frac{C}{|x|^{n-1}}$ when $|x|\gg 1$, this complete the proof.$\hfill\square$

Similarly, we have the following

Lemma 2**.**

[1, Lemma 2.5]** There exists a function $\phi_{0}(x)\in C^{2}(\Omega)\cap C(\overline{\Omega})$ satisfying the following boundary value problem

[TABLE]

Moreover, $\phi_{0}(x)$ satisfies:

$0<\phi_{0}(x)\leq C\ln(|x|)$, for all $x\in\Omega$.

• 2.

$\phi_{0}(x)\geq C$, for all $|x|\gg 1$.

• 3.

$|\nabla\phi_{0}(x)|\leq\frac{C}{|x|}$, for all $|x|\gg 1$.

Lemma 3**.**

[2, Lemma 2.2]** There exists a function $\phi_{0}(x)\in C^{2}([0,\infty))$ satisfying the following boundary value problem

[TABLE]

Moreover, $\phi_{0}(x)$ satisfies: there exist two positive constants $C_{1}$ and $C_{2}$ such that, for all $x>0$, we have $C_{1}x\leq\phi_{0}(x)\leq C_{2}x$. In fact, we can take $\phi_{0}(x)=Cx$.

3 Proof of Theorem 1

Proof of Theorem 1 1**.**

The idea to prove Theorem 1 is to use the variational formulation of the weak solution by choosing the appropriate test function. Note that the harmonic functions in Lemma 1, 2 and 3 play a crucial role in the exterior domain, because of their good behaviour and vanishing on the boundary $\partial\Omega$.

We argue by contradiction assuming that $u$ is not a blow-up solution of (1.1), we have

[TABLE]

for all $T>0$ and all compactly supported function $\varphi\in C^{2}([0,T]\times\Omega)$ such that $\varphi(\cdotp,T)=0$ and $\varphi_{t}(\cdotp,T)=0$. Take $\varphi(x,t)=\phi_{0}(x)\varphi^{\ell}_{T}(x)\eta_{T}^{k}(t)$ where $\phi_{0}$ is the harmonic function introduced in Lemma 1, 2 and 3, $\eta_{T}(t):=\eta(\frac{t^{2}}{T^{2}})$, $\ell,k\gg 1$, and $\eta(\cdotp)\in C^{\infty}(\mathbb{R}_{+})$ is a cut-off non-increasing function such that

[TABLE]

$0\leq\eta(t)\leq 1$* and $|\eta^{{}^{\prime}}(t)|\leq C$ for some $C>0$ and all $t>0$; and $\varphi_{T}(x)=\Phi(\frac{|x|}{T})$ with the following smooth, non-increasing cut-off function*

[TABLE]

such that $0\leq\Phi(r)\leq 1$, $|\Phi^{\prime}(r)|\leq C/r$ and $|\Phi^{\prime\prime}(r)|\leq C/r^{2}$. We obtain

[TABLE]

*where $\Omega_{1}:=\{x\in\Omega;\;|x|\leq 2T\}$. At this stage, we have to distinguishes three cases:

$\bullet$* Case 1: $n\geq 3$. To estimate the right-hand side of (1), we introduce the term $\varphi^{1/p}\varphi^{-1/p}$ in $I_{1}$, and we use Young’s inequality to obtain*

[TABLE]

On the other hand, using Lemma 1 with all properties of $\phi_{0}$, $T\gg 1$, and Young’s inequality, we conclude that

[TABLE]

where $\nabla\Omega_{1}:=\{x\in\Omega;\;T\leq|x|\leq 2T\}$. Similarly,

[TABLE]

Using (3.3)-(3.5), it follows from (1) that

[TABLE]

Now, we have to distinguishes 2 subcases.

$\bullet$* Case (i): $1<p<1+\frac{2}{n-1}$. By Lemma 1, we have: $|\nabla\phi_{0}(x)|\leq\frac{C}{|x|^{n-1}}\leq\frac{C}{T^{n-1}}\leq\frac{C}{T}$) in $\nabla\Omega_{1}$, therefore, using the change of variables: $y=T^{-1}x,$ $s=T^{-1}t$, we get from $(\ref{weaksolution3})$ that*

[TABLE]

where $C$ is independent of $T$. As $p<1+\frac{2}{n-1}\Longleftrightarrow-2p^{\prime}+1+n<0$, it follows, by letting $T\rightarrow\infty$ that

[TABLE]

contradiction.

$\bullet$* Case (ii): $p=1+\frac{2}{n-1}$. From (1) in the Case 1 and the fact that $p=1+\frac{2}{n-1},$ there exists a positive constant $D$ independent of $T$ such that*

[TABLE]

which implies that

[TABLE]

On the other hand, we use Hölder’s inequality instead of Young’s one in $I_{1}$, $I_{2}$, and $I_{3}$, together with the same change of variables, we get

[TABLE]

thanks to the fact that $p=1+\frac{2}{n-1}$. Similarly

[TABLE]

and

[TABLE]

Finally, using (3.9)-(3.11), it follows from (1) that

[TABLE]

hence, by letting $T\rightarrow\infty$ and using (3.8), we get a contradiction.

$\bullet$* Case 2: $n=2$. In this case, we have a blow-up result just in the sub-critical case ($1<p<1+\frac{2}{n-1}=3$). By repeating the same calculation in the Case of $n\geq 3$ and using Lemma 2 instead of Lemma 1 (noted that the big difference is the fact that $\phi_{0}(x)\leq C\ln(|x|)$), we easily conclude that*

[TABLE]

[TABLE]

and

[TABLE]

This implies that

[TABLE]

where we have used, e.g., the fact that $\ln(T)\leq C\,T^{p^{\prime}-3/2}$. By letting $T$ goes to infinity and using $p<{3}$, we obtain the desired contradiction.

$\bullet$* Case 3: $n=1$. For the case $1<p<3$, repeat the same calculation as in the Case of $n\geq 3$ and using Lemma 3 instead of Lemma 1, we easily get*

[TABLE]

[TABLE]

and

[TABLE]

Using the change of variables: $y=T^{-\alpha}x,$ $s=T^{-1}t$, we get from $(\ref{weaksolution3})$ that

[TABLE]

which leads to a contradiction by letting $T\rightarrow\infty$.

For the critical case $p=3$, we get the contradiction by applying a similar calculation as in the case (ii) above by taking into account the support of $\nabla\varphi_{T}$, $\Delta\varphi_{T}$, and $\partial_{t}\eta_{T}$.

This completes the proof of Theorem 1. $\hfill\square$

Acknowledgements

The author would like to express sincere gratitude to Professor Ryo Ikehata for valuable discussion.