Lower bounds for blow-up time of a nonlinear viscoelastic wave equation

Abstract

This paper deals with the blow-up for a class of nonlinear viscoelastic wave equation. Under certain conditions on the data, we construct a lower bound for the blow-up time when blow-up occurs.

Introduction

In this paper, we study the blow-up solution for the following nonlinear viscoelastic wave equation:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}-\Delta u+ \int^{t}_{0}g(t-s)\triangle u(s)\,ds+|u_{t}|^{q-2}u_{t}=|u|^{p-2}u, &(x,t)\in\Omega\times (0,T),\\ u(x,t)=0, &(x,t)\in\partial\Omega\times(0,T),\\ u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x), &x\in\Omega, \end{cases}\displaystyle \end{aligned}$$

(1.1)

where Ω is a bounded domain in $\mathbb{R}^{n}$ with a smooth boundary ∂Ω, g is a positive function satisfying some conditions to be specified later, and

$$\begin{aligned} 2< p, q\leq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty, & \mbox{if } n=1,2, \\ \frac{2n-2}{n-2}, & \mbox{if } n\geq3. \end{array}\displaystyle \right . \end{aligned}$$

(1.2)

The blow-up properties of the solution to (1.1) has been studied by many authors (see [1–5]). For instance, Messaoudi [2] studied (1.1) and proved a blow-up result for solutions with negative initial energy if $p>q\geq2$ and a global result for $2\leq p\leq q$. This result has been later improved by the same author in [3] to accommodate certain solutions with positive initial energy. In [4], Song and Zhong considered (1.1) for strong damping $-\Delta u_{t}$ and proved a blow-up result for solutions with positive initial energy by using the ideas of the ‘potential well’ theory introduced by Payne and Sattinger [6]. Wang [5] has investigated a sufficient conditions of the initial data with arbitrarily positive initial energy such that the corresponding solution of (1.1) with $q=2$ blows up in finite time. For related results, we refer the reader to [7–10].

When blow-up occurs, the blow-up time $T^{*}$ cannot usually be computed exactly. It is therefore of great importance in practice to determine lower and upper bounds for $T^{*}$. The aim of this note is to derive a lower bound for $T^{*}$ when blow-up occurs. We point out that it is, in general, very hard to obtain a lower bound estimate for viscoelastic wave equation problems, for the method to estimate the derivative of the control functional in parabolic cases is no longer effective and the memory part makes it difficult to estimate the energy. Our method is based on a first-order differential inequality technique for a suitably defined auxiliary function and makes use of some Sobolev-type inequality.

Before stating our main result, let us recall some results on the local existence, uniqueness, and blow-up of the solution

Theorem 1.1

(see [3])

Let $(u_{0}(x),u_{1}(x) )\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ and p, q satisfy condition (1.2). Let $g\in C^{1}[0,\infty)$ be a non-negative and non-increasing function satisfying

$$ 1-\int_{0}^{\infty}g(s)\,ds=l>0. $$

(1.3)

Then problem (1.1) has a unique local solution

$$u\in C \bigl([0, T_{m}); H_{0}^{1}(\Omega) \bigr), \qquad u_{t} \in C \bigl([0,T_{m}); L^{2}( \Omega) \bigr)\cap L^{q} \bigl(\Omega\times (0,T_{m}) \bigr) $$

for some $T_{m}>0$.

Remark 1.1

Condition (1.3) is necessary to guarantee the hyperbolicity and well-posedness of system (1.1).

Let λ be the best constant of the Sobolev embedding $H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$ and $\beta=\lambda/l^{\frac{1}{2}}$. We set

$$\alpha=\beta^{-\frac{p}{p-2}}, \quad E_{1}= \biggl(\frac{1}{2}- \frac{1}{p} \biggr)\alpha^{2}. $$

Define the energy functional $E(t)$ associated to our system (1.1),

$$E(t)=\frac{1}{2}\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \frac{1}{2}\biggl(1-\int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t)\bigr\| ^{2}_{2} +\frac{1}{2}(g\circ\nabla u) (t)-\frac{1}{p}\|u\|_{p}^{p}, $$

where

$$g\circ w(t)=\int^{t}_{0}g(t-s)\bigl\| w(s)- w(t) \bigr\| _{2}^{2}\,ds. $$

Moreover, we assume that g satisfies

$$ \int_{0}^{\infty}g(s)\,ds< \frac{p^{2}-2p}{p^{2}-2p+1}. $$

(1.4)

Then we have the following blow-up result.

Theorem 1.2

(see [3])

Assume that p, q satisfy condition (1.2) and g satisfies (1.3) and (1.4). If $p>q$ and the initial data $(u_{0},u_{1})$ satisfies

$$E(0)>E_{1}, \qquad \|\nabla u_{0}\|_{2}>\alpha, $$

then any solution of (1.1) blows up in finite time.

The main result

In this section, we switch to discuss the lower bound of the blow-up time for the blow-up solution of (1.1). Before we state and prove our main result, we need the following lemma.

Lemma 2.1

Suppose that (1.2), (1.3), and (1.4) hold. Let u be a solution of (1.1). Then energy functional $E(t)$ is non-increasing, that is, $E'(t)\leq0$.

Proof

By multiplying (1.1) by $u_{t}$ and integrating over Ω, we obtain

$$\begin{aligned} &\frac{d}{dt} \biggl( \frac{1}{2}\|u_{t} \|_{2}^{2}+ \frac{1}{2}\|\nabla u\|_{2}^{2}- \frac{1}{p}\|u\|_{p}^{p} \biggr)-\int _{0}^{t} g(t-s)\int_{\Omega}\nabla u(s)\cdot\nabla u_{t}(t)\,dx\,ds \\ &\quad=-\int_{0}^{t} \bigl\| u_{t}(s)\bigr\| _{q}^{q} \end{aligned}$$

(2.1)

for any regular solution. This result remains valid for weak solutions by a simple density argument. For the last term on the left side of (2.1), we have

$$\begin{aligned} &\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t)\cdot\nabla u(s)\,dx \,ds \\ &\quad=\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t) \bigl(\nabla u(s)-\nabla u(t) \bigr)\,dx\,ds+\int_{0}^{t}g(t-s)\int _{\Omega}\nabla u_{t}(t)\cdot\nabla u(t)\,dx\,ds \\ &\quad=-\frac{1}{2}\int_{0}^{t}g(t-s) \frac{d}{dt}\int_{\Omega}\bigl|\nabla u(s)-\nabla u(t)\bigr|^{2}\,dx\,ds+\frac{1}{2}\int_{0}^{t}g(s) \frac{d}{dt}\int_{\Omega}\bigl|\nabla u(t)\bigr|^{2} \,dx\,ds \\ &\quad=\frac{1}{2}\frac{d}{dt} \biggl(\int_{0}^{t}g(s)\,ds \bigl\| \nabla u(t)\bigr\| ^{2}_{2}-(g\circ\nabla u) (t) \biggr)+ \frac{1}{2}\bigl(g'\circ\nabla u\bigr) (t)- \frac{1}{2}g(t)\bigl\| \nabla u(t)\bigr\| ^{2}_{2}. \end{aligned}$$

(2.2)

Inserting (2.2) into (2.1), we get

$$E'(t)=-\int_{0}^{t} \bigl\| u_{t}(s)\bigr\| _{q}^{q}+\frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t)-\frac{1}{2}g(t)\bigl\| \nabla u(t) \bigr\| ^{2}_{2}\leq0, $$

where we also use g being non-negative and non-increasing function. □

Theorem 2.2

Assume that the conditions in Theorem 1.2 hold. Let $u(x, t)$ be the solution of problem (1.1), which blows up at a finite time $T^{*}$. Then

$$T^{*}\geq\int^{\infty}_{F(0)} \frac{1}{C_{3}y^{k}+y+C_{4}}\,dy, $$

where the constants $C_{3}$, $C_{4}$, and the exponent k will be defined in (2.9), and $F(0)=\int_{\Omega}|u_{0}|^{p}\,dx$.

Proof

Define $F(t)=\int_{\Omega}|u(t)|^{p}\,dx$. Then

$$ F'(t)=p\int_{\Omega}|u|^{p-2}uu_{t}\,dx \leq \frac{p}{2} \biggl(\int_{\Omega}|u|^{2p-2}\,dx+ \int_{\Omega}|u_{t}|^{2}\,dx \biggr). $$

(2.3)

To estimate the first term on the right side of inequality (2.3), we consider the following two cases.

Case 1. $2< p\leq\frac{2n}{n-1}$. Let $\gamma=2p-2$, $\mu=n(p-2)$, $2^{*}=\frac{2n}{n-2}$. Applying Hölder’s inequality and the embedding inequality, we have

$$ \int_{\Omega}|u|^{\gamma}\,dx=\int_{\Omega}|u|^{\gamma\theta}|u|^{\gamma (1-\theta)}\,dx \leq \biggl(\int_{\Omega}|u|^{\mu}\,dx \biggr)^{\frac{\gamma\theta}{\mu}} \biggl(\int_{\Omega}|u|^{2^{*}}\,dx \biggr)^{\frac{\gamma(1-\theta)}{2^{*}}}, $$

where θ satisfies

$$ \frac{\gamma\theta}{\mu}+\frac{\gamma(1-\theta)}{2^{*}}=1. $$

A straightforward computation shows

$$\begin{aligned}& \theta=\frac{1-\frac{\gamma}{2^{*}}}{\frac{\gamma}{\mu}-\frac{\gamma }{2^{*}}}=\frac{\mu(2^{*}-\gamma)}{\gamma(2^{*}-\mu)}, \\& \frac{\gamma\theta}{\mu}=\frac{2^{*}-\gamma}{2^{*}-\mu}=\frac {2}{n},\qquad \frac{\gamma-\theta\gamma}{2^{*}}=1- \frac{2}{n}, \end{aligned}$$

and then we have

$$\begin{aligned} \|u\|_{\gamma}^{\gamma} \leq&\|u\|_{\mu}^{\gamma\theta} \|u\|_{2^{*}}^{\gamma(1-\theta)} =\|u\|_{\mu}^{\frac{2\mu}{n}}\|u \|_{2^{*}}^{2} \\ \leq& C_{\ast}^{2}\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr)\|u\| _{p}^{\frac{2\mu}{n}}\|\nabla u\|_{2}^{2} \\ \leq& C_{\ast}^{2}\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr) \bigl(\|u\| _{p}^{\frac{2\mu}{n}\cdot s}+\|\nabla u\|_{2}^{2t} \bigr) \\ \leq& C_{1}\bigl(\| u\|_{p}^{p}+\|\nabla u \|_{2}^{2}\bigr)^{k_{1}}, \end{aligned}$$

(2.4)

where we have used the Hölder inequality,

$$ \|u\|_{\mu}^{\frac{2\mu}{n}}\leq|\Omega|^{\frac{2(p-\mu)}{np}}\|u\| _{p}^{\frac{2\mu}{n}}\leq\bigl(1+|\Omega|^{\frac{2(p-\mu)}{np}}\bigr)\|u\| _{p}^{\frac{2\mu}{n}}, $$

and

$$ \|u\|_{2^{*}}\leq C_{\ast}\|\nabla u\|_{2}, $$

here $C_{\ast}$ is the best constant of the Sobolev embedding $H^{1}_{0}(\Omega)\hookrightarrow L^{2^{\ast}}(\Omega)$; $\frac {1}{s}+\frac{1}{t}=1$, letting $t=\frac{2\mu}{pn}s$, from which we can deduce $k_{1}=\frac{3p-4}{p}$, $C_{1}=C_{\ast}^{2}(1+|\Omega|^{\frac{2(p-\mu)}{np}})$.

Case 2. $\frac{2n}{n-1}< p\leq\frac{2(n-1)}{n-2}$. Following the lines of the proof of inequality (2.4), we have

$$\begin{aligned} \|u\|_{\gamma}^{\gamma}\leq|\Omega|^{1-\frac{\gamma}{2^{*}}}\|u \| ^{\gamma}_{2^{*}} \leq& C^{\gamma}_{\ast}\bigl(1+| \Omega|^{1-\frac{\gamma }{2^{*}}}\bigr)\|\nabla u\|_{2}^{\gamma} \\ \leq&C^{\gamma}_{\ast}\bigl(1+|\Omega|^{1-\frac{\gamma}{2^{*}}}\bigr) \bigl(\|\nabla u\|_{2}^{\gamma}+\|u\|^{p(p-1)}_{p} \bigr) \\ \leq& C_{2}\bigl(\| u\|_{p}^{p}+\|\nabla u \|_{2}^{2}\bigr)^{k_{2}}, \end{aligned}$$

(2.5)

with $k_{2}=p-1$, $C_{2}=C^{\gamma}_{\ast}(1+|\Omega|^{1-\frac{\gamma }{2^{*}}})$.

From Lemma 2.1, we have

$$ E(t)\leq E(0)=\frac{1}{2}\| u_{1} \|^{2}_{2}+\frac{1}{2}\|\nabla u_{0} \|^{2}_{2}-\frac{1}{p}\| u_{0} \|^{p}_{p} = E_{0},\quad t\in\bigl[0,T^{*}\bigr). $$

(2.6)

Recalling the definition of $E(t)$, (1.4), and (2.6), we have

$$\begin{aligned} &\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \frac{1}{(p-1)^{2}}\bigl\| \nabla u(t)\bigr\| ^{2}_{2}+g\circ u(t) \\ &\quad\leq\bigl\| u_{t}(t)\bigr\| ^{2}_{2}+ \biggl(1-\int ^{t}_{0}g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2}_{2}+g\circ u(t) \\ &\quad= \frac{2}{p}\bigl\| u(t)\bigr\| ^{p}_{p}+2E(t)\leq \frac {2}{p}F(t)+2E_{0}. \end{aligned}$$

(2.7)

Combining (2.3)-(2.7), we get

$$\begin{aligned} F'(t) \leq&\frac{p}{2} \biggl(C_{i} \bigl(\| u\|_{p}^{p}+\|\nabla u\| _{2}^{2} \bigr)^{k_{i}}+\frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{p}{2} \biggl(C_{i} \biggl(F(t)+C_{0} \biggl(\frac{2}{p}F(t)+2E_{0}\biggr) \biggr)^{k_{i}}+ \frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{p}{2} \biggl(C_{i} \biggl(\biggl(1+ \frac{2C_{0}}{p}\biggr)F(t)+2C_{0}E_{0} \biggr)^{k_{i}}+\frac{2}{p}F(t)+2E_{0} \biggr) \\ \leq&\frac{pC_{i}}{2}2^{k_{i}-1} \biggl(\biggl(1+\frac {2C_{0}}{p} \biggr)^{k_{i}}F(t)^{k_{i}}+(2C_{0}E_{0})^{k_{i}} \biggr) +F(t)+pE_{0} \\ =&C_{3}F(t)^{k_{i}}+F(t)+C_{4}, \end{aligned}$$

(2.8)

where

$$\begin{aligned} &C_{0}=\frac{1}{1-\int^{\infty}_{0}g(s)\,ds}, \\ &C_{3}=\frac{pC_{i}2^{k_{i}}}{4}\biggl(1+\frac{2C_{0}}{p} \biggr)^{k_{i}}, \\ &C_{4}=pE_{0}+\frac{pC_{i}}{4}(4C_{0}E_{0})^{k_{i}}, \quad i=1,2. \end{aligned}$$

(2.9)

Applying Theorem 1.2, we have

$$ \lim_{t\rightarrow T^{*}}\int_{\Omega}|u|^{p}\,dx=+ \infty. $$

(2.10)

According to (2.8) and (2.10), we obtain

$$ \int^{\infty}_{F(0)}\frac{1}{C_{3}y^{k}+y+C_{4}}\,dy\leq T^{*}. $$

 □