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Stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms

Boundary Value Problems20182018:14

https://doi.org/10.1186/s13661-018-0932-z

  • Received: 16 August 2017
  • Accepted: 15 January 2018
  • Published:

Abstract

The goal of this study is to investigate an initial boundary value problem for the stochastic quasilinear viscoelastic wave equation involving the nonlinear damping \(\vert u_{t} \vert ^{q-2} u_{t}\) and a source term of the type \(\vert u \vert ^{p-2}u\) driven by additive noise. By an appropriate energy inequality, we prove that finite time blow-up is possible for equation (1.1) below if \(p > \{q, \rho +2 \}\) and the initial data are large enough (that is, if the initial energy is sufficiently negative). Also, we show that if \(q \geq p\), the local solution can be extended for all time and is thus global.

Keywords

  • stochastic quasilinear viscoelastic wave equation
  • blow-up of solution
  • global existence

MSC

  • 60H15
  • 35L05
  • 35L70

1 Introduction

In this paper, we are concerned with the following stochastic viscoelastic wave equation:
$$\begin{aligned}& \begin{aligned} &\vert u_{t} \vert ^{\rho }u_{tt} -\Delta u -\Delta u_{tt}+ \int_{0}^{t}h(t- \tau )\Delta u(\tau )\,d\tau + \vert u_{t} \vert ^{q-2} u_{t} \\ &\quad =\vert u \vert ^{p-2}u + \epsilon \sigma (x,t) \partial_{t} W(x,t)\quad\mbox{in }D \times (0,T), \\ & u=0\quad \mbox{on }\partial D \times (0,T), \\ & u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x)\quad\mbox{in }\bar{D}, \end{aligned} \end{aligned}$$
(1.1)
where D is a bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂D, with given positive constants \(\rho >0\), \(q \geq 2\), and \(p \geq 2\). The function \(h:\mathbb{R}^{+} \to \mathbb{R}^{+}\) in the viscoelastic term is a positive relaxation function satisfying some conditions to be specified later. \(W(x,t)\) is an infinite dimensional Wiener process, \(\sigma (x,t)\) is \(L^{2}(D)\)-valued progressively measurable, and ϵ is a given positive constant which measures the strength of noise.
System (1.1) without the stochastic term is a model for quasilinear viscoelastic wave equation with nonlinear damping and source terms. Various forms of the deterministic system (1.1) have been considered by many authors, and several results considering existence, nonexistence, and asymptotic behavior have been established in [15], and the references therein. For example, Liu [3] considered the following quasilinear viscoelastic wave equation problem:
$$\begin{aligned}& \vert u_{t} \vert ^{\rho }u_{tt} -\Delta u -\Delta u_{tt} + \int_{0}^{t}g(t- \tau )\Delta u(\tau )\,d\tau =b \vert u \vert ^{p-2}u\quad \mbox{in } D \times (0,\infty ), \\& u=0\quad\mbox{on } \partial D \times (0,\infty ), \\& u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x)\quad\mbox{in } \Omega , \end{aligned}$$
where D is a bounded domain in \(\mathbb{R}^{n}\) \((n \geq 1)\) with a smooth boundary ∂D, and ρ, \(b > 0\), \(p > 2\) are constants. The author investigated the general solution and blow-up solutions for this problem. Also, Song [4] studied the nonlinear quasilinear viscoelastic wave equation problem
$$\begin{aligned}& \vert u_{t} \vert ^{\rho }u_{tt} -\Delta u + \int_{0}^{t}g(t-\tau )\Delta u( \tau )\,d\tau + \vert u_{t} \vert ^{m-2}u_{t} =\vert u \vert ^{p-2}u \quad\mbox{in } D \times [0,T], \\& u=0 \quad\mbox{on }\partial D \times [0,T], \\& u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x)\quad \mbox{in }D, \end{aligned}$$
where D is a bounded domain of \(\mathbb{R}^{n}\) \((n \geq 1)\) with a smooth boundary ∂D, \(m>2\), \(g:\mathbb{R}^{+} \to \mathbb{R} ^{+}\) is a positive nonincreasing function, and
$$\begin{aligned}& 2< p< \infty \quad \mbox{if }n=1,2,\qquad 2< p< 2(n-1)/(n-2) \quad\mbox{if }n\geq 3, \\& 2< \rho < \infty \quad\mbox{if }n=1,2,\qquad 2< \rho \leq n/(n-2) \quad\mbox{if }n \geq 3. \end{aligned}$$
He proved the global nonexistence of positive initial energy solutions for a quasilinear viscoelastic wave equation.

Under the consideration of random environment, there are many studies on the stochastic wave equation with global existence and invariant measures for linear and nonlinear damping (see the references in [625]).

Wei and Jiang [26] and Gao, Guo and Liang [24] considered the following nonlinear stochastic viscoelastic wave equation:
$$\begin{aligned}& u_{tt} -\Delta u + \int_{0}^{t}h(t-\tau )\Delta u(\tau )\,d\tau + u _{t} \\& \quad =\vert u \vert ^{p-2}u + \epsilon \sigma (u, \nabla u, x,t) \partial_{t} W(x,t) \quad\mbox{in } D \times (0,T), \\& u=0 \quad \mbox{on } \partial D \times (0,T), \\& u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x)\quad\mbox{in }\bar{D}. \end{aligned}$$
They investigated the global existence and the energy decay estimate of a solution and showed that the solution blows up with positive probability or it is explosive in \(L^{2}\) sense under some conditions.

Moreover, Cheng et al. [23] proved the existence of a global solution and blow-up solutions with positive probability for the nonlinear stochastic viscoelastic wave equation with linear damping (see [18, 22, 26]).

Recently, Cheng et al. [23] studied the stochastic viscoelastic wave equation with nonlinear damping and source terms
$$\begin{aligned}& u_{tt} -\Delta u + \int_{0}^{t}h(t-\tau )\Delta u(\tau )\,d\tau + \vert u_{t} \vert ^{q-2} u_{t} \\& \quad = \vert u \vert ^{p-2}u + \epsilon \sigma (x,t) \partial_{t} W(x,t)\quad \mbox{in }D \times (0,T), \\& u=0\quad\mbox{on }\partial D \times (0,T), \\& u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x)\quad \mbox{in }\bar{D}, \end{aligned}$$
where D is a bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂D, \(q\geq 2, p \geq 2\), ϵ is a given positive constant which measures the strength of noise; \(W(x,t)\) is an infinite dimensional Wiener process; \(\sigma (x,t,w)\) is \(L^{2}(D)\)-valued progressively measurable; and h is a positive relaxation function. The authors studied the global solution of stochastic viscoelastic wave equations with nonlinear damping and source terms.

The previous work in Cheng et al. [23] established that the solution blows up with positive probability or it is explosive in energy sense for \(p>q\). Motivated by this work, we prove that the stochastic quasilinear viscoelastic wave equation (1.1) can blow up with positive probability or it is explosive in energy sense for \(p > \{q, \rho +2 \}\) and obtain the existence of global solution by the Borel-Cantelli lemma. To the best of our knowledge, there have been no results for the blow-up of solutions of stochastic quasilinear viscoelastic wave equation with positive probability.

This paper is organized as follows. In Section 2, we present some assumptions, definitions, and lemmas needed for our work. The result for the local existence and a pointwise unique solution of equation (1.1) are given too. In Section 3, we show Lemmas 3.1 and 3.2. With those lemmas, we prove our main result for \(p> \{q, \rho +2 \}\). In Section 4, we obtain global existence of equation (1.1).

2 Preliminaries

Let \((X,\Vert \cdot \Vert _{X} )\) be a separable Hilbert space with Borel σ-algebra \(\mathbf{B}(X)\), and let \((\Omega ,\mathfrak{F}, P)\) be a probability space. We set \(H=L^{2}(D)\) with the inner product and norm denoted by \(( \cdot , \cdot )\) and \(\Vert \cdot \Vert \), respectively. We denote by \(\Vert \cdot \Vert _{q}\) the \(L^{q} (D)\) norm for \(0 \leq q \leq \infty \) and by \(\Vert \nabla \cdot \Vert \) the Dirichlet norm in \(V=H_{0}^{1}(D)\) which is equivalent to \(H^{1}(D)\) norm.

First, we introduce the following hypotheses:
  1. (H1)
    We assume that p, q, ρ satisfy
    $$\begin{aligned} \begin{aligned} &q \geq 2, \qquad p>2,\qquad \max \{p,q \} \leq \frac{2(n-1)}{n-2} \quad \mbox{if } n\geq 3; \\ & q \geq 2, \qquad p>2 \quad \mbox{if } n=1,2; \\ & 0\leq \rho \leq \frac{2}{n-2} \quad \mbox{if } n\geq 3, \qquad 0< \rho < \infty \quad \mbox{if } n=1,2. \end{aligned} \end{aligned}$$
    (2.1)
     
  2. (H2)
    We assume that \(h:\mathbb{R}^{+} \to \mathbb{R} ^{+}\) is a bounded nonincreasing \(C^{1}\) function satisfying
    $$ h(s)>0, \quad 1-\int_{0}^{\infty } h(s)\,ds =l>0, $$
    and there exist positive constants \(\xi_{1}\) and \(\xi_{2}\) such that
    $$\begin{aligned} -\xi_{1} h(t) \leq h^{\prime }(t) \leq - \xi_{2} h(t), \quad t \geq 0. \end{aligned}$$
    (2.2)
     
  3. (H3)
    \(\sigma (x,t)\) is \(H_{0}^{1}(D) \cap L^{\infty }(D)\)-valued progressively measurable such that
    $$\begin{aligned} E \int_{0}^{T} \bigl(\bigl\Vert \nabla \sigma (t) \bigr\Vert ^{2} +\bigl\Vert \sigma (t) \bigr\Vert _{\infty } ^{2} \bigr)\,dt < \infty . \end{aligned}$$
    (2.3)
     

Lemma 2.1

([8])

For all \(u,v \in H^{1}(\mathbb{R}^{n})\) and \(0< p \leq \frac{2}{n-2}\) (\(n\geq 3\)) or \(p > 0\) (\(n=1,2\)), there exists a constant \(c_{1}=c_{1} (n,p)>0\) such that
$$ \Vert u \Vert _{L^{2(p-1)}} \leq c_{1} \Vert u \Vert _{H^{1}}, \qquad \bigl\Vert u^{p} v \bigr\Vert \leq c _{1}^{p+1} \Vert u \Vert _{H^{1}}^{p} \Vert v \Vert _{H^{1}}. $$
(2.4)
In this paper, \(E(\cdot )\) stands for expectation with respect to probability measure P, and \(W(x,t)(t\geq 0)\) is a V-valued Q-Wiener process on the probability space with the covariance operator Q satisfying \(\operatorname{Tr}(Q)<\infty \). A complete orthonormal system \(\{e_{k}\}_{k=1}^{\infty }\) in V with \(c_{0} : =\sup_{k\geq 1}\Vert e_{k} \Vert _{\infty } <\infty \) and a bounded sequence of nonnegative real members \(\{ \lambda_{k} \}_{k=1}^{\infty }\) satisfy that
$$\begin{aligned} Qe_{k} = \lambda_{k} e_{k}, \quad k=1,2,\ldots . \end{aligned}$$
To simplify the computations, we assume that the covariance operator Q and Laplacian − with a homogeneous Dirichlet boundary condition have a common set of eigenfunctions, that is,
$$\begin{aligned}& -\triangle e_{k} = \mu_{k} e_{k}, \quad x\in D, \\& e_{k} =0, \quad x\in \partial D, \end{aligned}$$
and then, for any \(t \in [0,T]\), \(W(x,t)\) has an expansion
$$\begin{aligned} W(x,t)=\sum_{k=1}^{\infty } \sqrt{ \lambda_{k}}\beta_{k}(t) e_{k}(t), \end{aligned}$$
(2.5)
where \(\{\beta_{k}(t)\}_{k=1}^{\infty }\) are real-valued Brownian motions mutually independent of \((\Omega , \mathfrak{F},P)\). Let \(\mathcal{H}\) be the set of \(L_{2}^{0} =L^{2}(Q^{1/2}V,V)\)-valued processes with the norm
$$\begin{aligned} \bigl\Vert \Phi (t) \bigr\Vert _{\mathcal{H}}=\biggl(E \int_{0}^{t}\bigl\Vert \Phi (s) \bigr\Vert _{L_{2}^{0}} ^{2}\,ds\biggr)^{1/2}= \biggl(E \int_{0}^{t} \operatorname{Tr}\bigl(\Phi (s) Q \Phi^{*}(s)\bigr)\,ds\biggr)^{1/2} < \infty , \end{aligned}$$
(2.6)
where \(\Phi^{*}(s)\) denotes the adjoint operator of \(\Phi (s)\). For any \(\Phi^{*}(t) \in \mathcal{H}\), we can define the stochastic integral with respect to the Q-Wiener process as \(\int_{0}^{t} \Phi (s)\,dW(s)\), which is martingale. For more details about the finite dimension Winner process and the stochastic integral, see [22].

Definition 2.1

Assume that \((u_{0},u_{1}) \in H_{0}^{1}(D) \times L^{2}(D)\) and \(E\int_{0}^{T}\Vert \sigma (t) \Vert ^{2}\,dt < \infty \). u is said to be the solution of (1.1) on the interval \([0,T]\) if \((u,u_{t})\) is \(H_{0}^{1}(D) \times L^{2}(D)\)-valued progressively measurable, \((u,u_{t})\in L^{2}(\Omega ; C([0,T];H_{0}^{1}(D) \times L^{2}(D)))\), \(u_{t} \in L^{q}((0,T)\times D)\), and such that (1.1) holds in the sense of distributions over \((0,T)\times D\) for almost all w.

By combining the arguments of [20, 23, 24], we get the existence result.

Theorem 2.1

([20, 24])

Assume that (H1)-(H3) hold. Then, for the initial data \((u_{0},u_{1}) \in (H^{2}(D) \cap H_{0} ^{1}(D))\times H_{0}^{1}(D)\), problem (1.1) has a pointwise unique solution u such that
$$ u \in L^{2}\bigl(\Omega ; L^{\infty }\bigl(0,T; H^{2}(D) \cap H_{0}^{1}(D)\bigr)\bigr) \cap L^{2}\bigl(\Omega ; C\bigl([0,T];H_{0}^{1}(D) \bigr)\bigr) $$
and
$$ u_{t} \in L^{2}\bigl(\Omega ; L^{\infty } \bigl(0,T;H_{0}^{1}(D)\bigr)\bigr)\cap L^{2}\bigl( \Omega ;C\bigl([0,T];L^{2}(D)\bigr)\bigr). $$

3 Blow-up result

In this section, we prove our main result for \(p>q\). For this purpose, we give defined restrictions on \(\sigma (x,t)\) and the relaxation function h such that
$$\begin{aligned} E \int_{0}^{\infty } \int_{D} \sigma^{2}(x,t)\,dx\,dt < \infty , \qquad \int_{0}^{\infty }h(s)\,ds < \frac{{p(p-2)}}{{(p-1)^{2}}}. \end{aligned}$$
(3.1)
Now, we define an energy function
$$\begin{aligned} F(t)&=\frac{{1}}{{\rho +2}}\bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{{1}}{{2}}\bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2} \\ &\quad {} +\frac{{1}}{{2}}\biggl(1- \int_{0}^{t} h(s)\,ds\biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{{1}}{{2}}(h \circ \nabla u) (t) - \frac{{1}}{{p}}\bigl\Vert u(t) \bigr\Vert _{p}^{p}, \end{aligned}$$
(3.2)
where
$$\begin{aligned} (h \circ \nabla u) (t)= \int_{0}^{t} h(t-s)\bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
For each N, stopping time \(\tau_{N}\) is given as
$$ \tau_{N}=\inf \bigl\{ t>0: \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \geq N \bigr\} , $$
where \(\tau_{N}\) is increasing in N, and \(\tau_{\infty }= \lim_{N \to \infty } \tau_{N}\). In order to prove our blow-up result, we rewrite (1.1) as an equivalent Itô’s system
$$\begin{aligned}& \begin{aligned} &du=v\,dt, \\ &d\biggl(\frac{{1}}{{\rho +1}} \vert v \vert ^{\rho }v-\Delta v\biggr)= \biggl(\Delta u -\int_{0} ^{t} h(t-s) \Delta u(s)\,ds -\vert v \vert ^{q-2}v +\vert u \vert ^{p-2}u\biggr)\,dt \\ &\hphantom{\biggl(\frac{{1}}{{\rho +1}} \vert v \vert ^{\rho }v-\Delta v\biggr)=\quad }{}+\epsilon \sigma (x,t)\,dW_{t}(x,t), \quad (x,t)\in D \times (0,T), \\ & u(x,t)=0, \quad (x,t) \in \partial D \times (0,T), \\ & u(x,0)=u_{0}(x), \qquad v(x,0)=v_{0}(x)=u_{1}(x), \quad x\in D, \end{aligned} \end{aligned}$$
(3.3)
where \((u_{0},u_{1})\in H_{0}^{1}(D)\times L^{2}(D)\). Then the energy function \(F(t)\) becomes
$$\begin{aligned} F(t)&=\frac{{1}}{{\rho +2}}\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{{1}}{{2}}\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {} +\frac{{1}}{{2}}\biggl(1- \int_{0}^{t} h(s)\,ds\biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{{1}}{{2}}(h \circ \nabla u) (t) - \frac{{1}}{{p}}\bigl\Vert u(t) \bigr\Vert _{p}^{p}. \end{aligned}$$
(3.4)

Lemma 3.1

Let \((u,v)\) be a solution of Eq. (3.3) with the initial data \((u_{0},v_{0})\in H_{0}^{1}(D)\times L^{2}(D)\). Then we have
$$\begin{aligned} \frac{{d}}{{dt}}E F(t)&= -E \bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\epsilon ^{2}}}{{2}} \sum_{j=1}^{\infty } E\int_{D} \lambda _{j} e_{j}^{2}(x) \sigma^{2}(x,t)\,dx \\ & \quad {}-E\bigl(-h'\circ \nabla u\bigr) (t)-\frac{{1}}{{2}}h(t)E \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \leq -E \bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\epsilon ^{2}}}{{2}} \sum_{j=1} ^{\infty } E \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,t)\,dx, \end{aligned}$$
(3.5)
and
$$\begin{aligned} &E\biggl(u(t),\frac{{1}}{{\rho +1}}\bigl\vert v(t) \bigr\vert ^{\rho }v(t) - \Delta v(t)\biggr) \\ &\quad =\biggl(u_{0}, \frac{{1}}{{\rho +1}}\vert u_{1} \vert ^{\rho }u_{1}- \Delta u_{1}\biggr)- \int_{0}^{t} E\bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds \\ & \quad \quad {} - \int_{0}^{t} E\bigl(u(s),\bigl\vert v(s) \bigr\vert ^{q-2}v(s)\bigr)\,ds + \int_{0}^{t} E \bigl\Vert u(s) \bigr\Vert _{p} ^{p}\,ds \\ &\quad \quad {} +E \int_{0}^{t} \int_{0}^{s} h(s-\tau ) \bigl(\nabla u(\tau ), \nabla u(s)\bigr)\,d\tau \,ds \\ &\quad \quad {} +\frac{{1}}{{\rho +1}}E \int_{0}^{t} \bigl\Vert v(s) \bigr\Vert _{\rho +2}^{\rho +2}\,ds +E \int_{0}^{t} \bigl\Vert \nabla v(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.6)

Proof

By multiplying Eq. (3.3) by \(v(t)\) and using Itô’s formula, we deduce (3.5). Also, multiplying Eq. (3.3) by \(u(t)\) and integrating by parts over \((0,T)\), we arrive at (3.6) (see [24]). □

Let
$$ G(t) =\frac{{\epsilon ^{2}}}{{2}}\sum_{j=1}^{\infty } E \int_{0}^{t} \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,s)\,dx \,ds. $$
(3.7)
Due to (3.1), we deduce
$$\begin{aligned} G(\infty )&= \frac{{\epsilon ^{2}}}{{2}}\sum_{j=1}^{\infty } E \int _{0}^{\infty } \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,s)\,dx \,ds \\ & \leq \frac{{\epsilon ^{2}}}{{2}} \operatorname{Tr}(Q) c_{0}^{2} E \int_{0}^{\infty } \int_{D} \sigma^{2}(x,s)\,dx \,ds=E_{1} < \infty . \end{aligned}$$
(3.8)
We set
$$ H(t)=G(t)-E\bigl[F(t)\bigr]. $$
Then (3.5) implies that
$$ H^{\prime }(t)=G^{\prime }(t) - \frac{{d}}{{dt}}E \bigl[F(t)\bigr] \geq E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} \geq 0. $$
(3.9)

Lemma 3.2

Let \((u,v)\) be a solution of Eq. (3.3). Then there exists a positive constant C such that
$$\begin{aligned} E\bigl\Vert u(t) \bigr\Vert _{p}^{s} & \leq C\bigl[ G(t)-H(t) -E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} -E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} +E\bigl\Vert u(t) \bigr\Vert _{p}^{p} - E(h \circ \nabla u) (t) \bigr], \quad 2\leq s\leq p. \end{aligned}$$
(3.10)

Proof

If \(\Vert u \Vert _{p} \leq 1\), then \(\Vert u \Vert _{p}^{s} \leq \Vert u \Vert _{p}^{2} \leq C\Vert \nabla u \Vert ^{2}\) by the Sobolev embedding theorem. If \(\Vert u \Vert _{p} \geq 1\), then \(\Vert u \Vert _{p}^{s} \leq \Vert u \Vert _{p}^{p}\). Thus there exists a constant \(C>0\) such that \(E\Vert u \Vert _{p}^{s} \leq C( E\Vert \nabla u \Vert ^{2} + E\Vert u \Vert _{p}^{p})\). Therefore, combining with the definition of energy function, we get (3.10). □

Theorem 3.1

Assume that (H1)-(H3) and (3.1) hold. Let \((u,v)\) be a solution of Eq. (3.3) with the initial data \((u_{0},v _{0}) \in H_{0}^{1}(D)\times L^{2}(D)\) satisfying
$$\begin{aligned} F(0) \leq -(1+\beta )E_{1}, \end{aligned}$$
(3.11)
where \(\beta >0\) is an arbitrary constant and \(E_{1}\) is defined in (3.8). If \(p> \{ q, \rho +2 \}\), then the solution \((u,v)\) and the lifespan \(\tau_{\infty }\) defined above are either
  1. (1)

    \(P(\tau_{\infty }<\infty ) > 0\), that is, \(\Vert \nabla u(t) \Vert \) blows up in finite time with positive probability, or

     
  2. (2)
    there exists a positive time \(T^{*} \in [0,T_{0}]\) such that
    $$ \lim_{t \to T^{*} } E\bigl[F(t)\bigr]= +\infty , $$
    (3.12)
    where
    $$\begin{aligned} \begin{aligned} & T_{0} = \frac{1-\alpha }{\alpha K L^{\alpha /(1-\alpha )}(0)} , \\ &L(0)=H^{1-\alpha }(0) + \delta E\biggl(u_{0}, \frac{1}{\rho +1} \vert u_{1} \vert ^{ \rho } u_{1} - \Delta u_{1} \biggr) >0, \end{aligned} \end{aligned}$$
    (3.13)
    and α, K are given later.
     

Proof

For the lifespan \(\tau_{\infty }\) of the solution \(\{ u(t): t>0 \}\) of Eq. (3.3) with \(H_{0}^{1}(D)\) norm, we treat the case when \(P( \tau_{\infty }= +\infty ) <1\). Then, for sufficiently large \(T>0\), by (3.9) and (3.11), we obtain
$$\begin{aligned} 0< (1+\beta )E_{1} \leq -F(0)=H(0)\leq H(t)\leq G(t) +\frac{1}{p}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \leq E_{1} + \frac{1}{p}E\bigl\Vert u(t) \bigr\Vert _{p} ^{p}. \end{aligned}$$
(3.14)
Define
$$ L(t)=H^{1-\alpha }(t) + \delta E\biggl(u(t), \frac{1}{\rho +1} \bigl\vert v(t) \bigr\vert ^{ \rho }v(t) - \Delta v(t) \biggr), $$
where
$$ 0< \alpha < \min \biggl\{ \frac{1}{2}, \frac{p-2}{2p}, \frac{p-q}{pq}, \frac{1}{ \rho +2}-\frac{1}{p}\biggr\} $$
(3.15)
and δ is a very small constant to be determined later.
Using (3.6) and (3.9), we deduce
$$\begin{aligned} L^{\prime }(t)& =(1-\alpha ) H^{-\alpha }(t)H^{\prime }(t) +\delta \biggl[ -E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} -E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2} v(t)\bigr) \\ & \quad {}+ E \bigl\Vert u(t) \bigr\Vert _{p}^{p} +E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u( \tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad {}+ \frac{{1} }{{\rho +1}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \biggr] \\ &\geq (1-\alpha ) H^{-\alpha }(t) E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p \bigl[H(t) -G(t) + E F(t)\bigr] \\ & \quad {}-\delta E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} -\delta E \bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) + \delta E \bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ & \quad {} +\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau + \frac{{\delta } }{{\rho +1}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \delta E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ & \geq (1-\alpha ) H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p H(t) \\ & \quad {}+ \delta \biggl(\frac{{p} }{{\rho +2}}+ \frac{{1} }{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ & \quad {}+ \delta \biggl( \frac{p}{2} -1\biggr) E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \quad {}+\delta \biggl( \frac{{p} }{{2}}+1\biggr)E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) \\ & \quad {}+\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad {}+\frac{{\delta p}}{{2}} E(h \circ \nabla u) (t)- \frac{{\delta p}}{{2}}E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2} -\delta p G(t). \end{aligned}$$
(3.16)
On the other hand, we have
$$\begin{aligned} & \delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad =\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ) - \nabla u(t) ,\nabla u(t)\bigr)\,d\tau \\ & \qquad {}+\delta E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.17)
and by Hölder’s inequality, we get
$$\begin{aligned} & \delta E\int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ) - \nabla u(t) , \nabla u(t)\bigr)\,d\tau \\ & \quad \geq -\frac{{\delta p}}{{2}} E(h \circ \nabla u) (t) - \frac{{\delta }}{{2p}} E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.18)
Inserting (3.17) and (3.18) into (3.16), we obtain
$$\begin{aligned} L^{\prime }(t) &\geq (1-\alpha )H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) \\ &\quad {}-\delta p G(t) +\delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) E \int_{0}^{t} h( \tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.19)
For \(q< p\), by \(E\Vert u(t) \Vert _{q}^{q} \leq cE\Vert u(t) \Vert _{p}^{q}\) and Hölder’s inequality, we deduce the following estimate (see [23]):
$$\begin{aligned} E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t) \bigr)& \leq \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{ \frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{1}}{{q}}} \\ & \leq C\bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{q} \bigr)^{ \frac{{1}}{{q}}} \\ & \leq C \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{p}}} \\ & \leq C\bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{\frac{{1}}{{p}} -\frac{{1}}{{q}}} \end{aligned}$$
(3.20)
and Young’s inequality
$$\begin{aligned} & \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{q}}} \leq \frac{{q-1}}{{q}}\mu E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\mu ^{1-q}}}{{q}}E\bigl\Vert u(t) \bigr\Vert _{p}^{p}, \end{aligned}$$
(3.21)
where μ is a constant to be determined later. In view of (3.14), we get
$$ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \geq p\bigl(H(t)-G(t)\bigr)\geq \widetilde{\rho } H(t), $$
(3.22)
where \(\widetilde{\rho } = p\beta /(1+\beta )\). With the assumption of \(H(0)>1\), (3.21), (3.22), and (3.15) imply that
$$ \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{\frac{{1}}{{p}} -\frac{{1}}{{q}}} \leq \widetilde{\rho }^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}H(t)^{ \frac{{1}}{{p}} -\frac{{1}}{{q}}} \leq \widetilde{\rho }^{ \frac{{1}}{{p}} -\frac{{1}}{{q}}}H^{-\alpha }(t) \leq \widetilde{\rho } ^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}H^{-\alpha }(0). $$
(3.23)
Combining with (3.20), (3.21), and (3.23), we arrive at
$$ \bigl\vert E\bigl(u(t), \bigr\vert v(t)\bigl\vert ^{q-2}v(t)\bigr) \bigr\vert \leq a_{1} \frac{{q-1}}{{q}}\mu E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} H^{-\alpha }(t) + a_{1} \frac{{\mu ^{1-q}}}{{q}}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} H^{-\alpha }(t), $$
(3.24)
where \(a_{1}=C\widetilde{\rho }^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}\). Hence, substituting (3.24) for (3.19), we get
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} +\delta p H(t) \\ &\quad {}+ \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {}+ \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}+ \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta p G(t) \\ &\quad {}+ \delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h(\tau )\,d\tau E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}-\delta a_{1} \frac{ \mu^{1-q}}{q} E\bigl\Vert u(t) \bigr\Vert _{p}^{p} H^{- \alpha } (0). \end{aligned}$$
(3.25)
Using Lemma 3.2 with \(s=p\) and (3.25), we have
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} +\delta p H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta p G(t) \\ &\quad {} +\delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}-\delta a_{2} \mu^{1-q} \bigl[G(t) -H(t) - E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} -E\bigl\Vert \nabla v(t) \bigr\Vert ^{2}+E\bigl\Vert u(t) \bigr\Vert _{p}^{p} -E(h\circ \nabla u) (t)\bigr] \\ &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} \\ &\quad {}+\delta \bigl( p +a_{2} \mu^{1-q}\bigr) H(t) -\delta \bigl( p+ a_{2} \mu^{1-q}\bigr)G(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}+ a_{2} \mu ^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1 +a_{2} \mu^{1-q} \biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} - \delta a_{2} \mu^{1-q}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ &\quad {} +\delta a_{2} \mu^{1-q}E(h\circ \nabla u) (t) \\ &\quad {}+\delta \biggl[\frac{{p}}{{2}}-1 +\biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h( \tau )\,d\tau \biggr]E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.26)
where \(a_{2}=Ca_{1} H^{-\alpha }(0)/q \).
Note that
$$\begin{aligned} H(t) &\geq G(t) +\frac{{1}}{{p}} E\bigl\Vert u(t) \bigr\Vert _{p}^{p} - \frac{{1}}{{\rho +2}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}-\frac{{1}}{{2}}E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\frac{1}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \frac{{1}}{{2}}E(h\circ \nabla u) (t) \end{aligned}$$
(3.27)
with
$$ a_{3}= \frac{{p}}{{2}}-1 +\biggl(1- \frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h( \tau )\,d\tau >0, $$
(3.28)
we write \(p=2a_{4}+(p-2a_{4})\) with \(a_{4}=\min \{ a_{1}, a_{3}\}\), then estimate (3.26) yields
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \\ & \quad {}+ \delta \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) H(t) - \delta \bigl( p-2a_{4} +a _{2} \mu^{1-q} \bigr) G(t) \\ & \quad {}+ \delta \biggl( \frac{{p}}{{\rho +2}}-\frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ & \quad {}+ \delta \biggl(\frac{{p}}{{2}}-a_{4}+1+a_{2}\mu^{1-q}\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ \delta \biggl(- a_{2} \mu^{1-q}+\frac{{2a_{4}}}{{p}}\biggr) E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ & \quad {}+ \delta \bigl( a_{2} \mu^{1-q}-a_{4}\bigr)E(h\circ \nabla u) (t) + \delta (a _{3}-a_{4})E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.29)
From (3.8) and (3.14), we deduce
$$\begin{aligned} \bigl( p-2a_{4} +a_{2} \mu^{1-q} \bigr) G(t) &\leq \bigl( p-2a_{4} +a_{2} \mu^{1-q} \bigr)E _{1} \\ & \leq \frac{{ p-2a_{4} +a_{2} \mu ^{1-q}}}{{1+\beta }} H(t) . \end{aligned}$$
(3.30)
Substituting (3.30) with (3.29), we get
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \\ &\quad {} + \delta \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) \frac{{\beta }}{{1+\beta }} H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}-\frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}-a_{4}+1 +a_{2} \mu^{1-q}\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(- a_{2} \mu^{1-q}+\frac{{2a_{4}}}{{p}} \biggr) E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ &\quad {} + \delta \bigl( a_{2} \mu^{1-q}-a_{4}\bigr)E(h \circ \nabla u) (t) + \delta (a _{3}-a_{4})E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.31)
Next, we can choose μ large enough so that (3.31) becomes
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta \gamma \bigl(H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ & \quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr), \end{aligned}$$
(3.32)
where
$$\begin{aligned} \gamma &= \min \biggl\{ \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) \frac{{\beta }}{{1+\beta }}, \frac{{p}}{{\rho +2}}- \frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}, \\ & \quad {} \frac{{p}}{{2}}-a_{4}+1+a_{2} \mu^{1-q}, - a_{2} \mu^{1-q}+ \frac{{2a_{4}}}{{p}}, a_{2} \mu^{1-q}-a_{4}, a_{3}-a_{4} \biggr\} >0. \end{aligned}$$
Once μ is fixed, we pick δ small enough so that
$$ 1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta >0. $$
Using this, (3.32) takes the form
$$\begin{aligned} L^{\prime }(t) &\geq \delta \gamma \bigl(H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{ \rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr). \end{aligned}$$
(3.33)
Thus, we see that
$$\begin{aligned} L(t) \geq L(0)=H^{1-\alpha }(0) + \delta E \biggl(u_{0}, \frac{{1}}{{\rho +1}}\vert u_{1} \vert ^{\rho }u_{1} - \Delta u_{1}\biggr)>0, \quad \forall t \geq0. \end{aligned}$$
(3.34)
Consequently, we get
$$ L(t) \geq L(0)>0, \quad \forall t \geq 0. $$
(3.35)
Since
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t) \,dx \biggr\vert &\leq E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{ \rho +1} E\bigl\Vert u(t) \bigr\Vert _{\rho +2} \\ & \leq CE\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +1} E\bigl\Vert u(t) \bigr\Vert _{p}, \end{aligned}$$
we have
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t)\,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} &\leq E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\frac{{\rho +1}}{{1-\alpha }}} E\bigl\Vert u(t) \bigr\Vert _{ \rho +2}^{\frac{{1}}{{1-\alpha }}} \\ &\leq C \bigl[\bigl(E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2}\bigr)^{ \frac{ {(\rho +1)}}{{(\rho +2)(1-\alpha )}}\zeta } + \bigl(E\bigl\Vert u(t) \bigr\Vert _{p} ^{2}\bigr)^{\frac{{\theta }}{{2(1-\alpha )}}}\bigr], \end{aligned}$$
(3.36)
where \(\frac{{1}}{{\zeta }} +\frac{{1}}{{\theta }}=1\). By choosing \(\zeta = \frac{{(1-\alpha )(\rho +2)}}{{\rho +1}} (>1)\), we have \(\frac{{\theta }}{{2(1-\alpha )}}= \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}} <\frac{{p}}{{2}}\). And with (3.15), (3.36) becomes
$$\begin{aligned} \biggl\vert E \int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t) \,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} \leq C \bigl\{ E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{2}\bigr)^{ \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}}}\bigr\} . \end{aligned}$$
Using Lemma 3.2 with \(s= \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}}\), we obtain
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t)\,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} & \leq C \bigl[ H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}\bigr] . \end{aligned}$$
Therefore, we deduce, for all \(t\geq 0\),
$$\begin{aligned} L^{\frac{{1}}{{1-\alpha }}}(t)&=\biggl(H^{1-\alpha }(t) + \frac{{\delta }}{{\rho +1}} E \int_{D} \bigl\vert v(t) \bigr\vert ^{\rho }v(t)u(t) \,dx +\delta E \int_{D} \nabla v(t) \cdot \nabla u(t)\,dx \biggr)^{ \frac{{1}}{{1-\alpha }}} \\ & \leq C \bigl[ H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}\bigr] . \end{aligned}$$
(3.37)
Combining (3.33) and (3.37)
$$\begin{aligned} L^{\prime }(t) \geq KL^{\frac{{1}}{{1-\alpha }}}(t),\quad \forall t\geq 0 \end{aligned}$$
with a positive constant K depending on C and δγ, it follows that
$$\begin{aligned} L^{\frac{{\alpha }}{{1-\alpha }}}(t) \geq \frac{{1-\alpha }}{{(1-\alpha )L^{-{{\alpha }\over {1-\alpha }}}(0)-\alpha K t }}. \end{aligned}$$
Let
$$\begin{aligned} T_{0} = \frac{{1-\alpha }}{{\alpha K L^{{{\alpha }\over {1-\alpha }}}(0)}}. \end{aligned}$$
Then \(L(t) \to \infty \) as \(t \to T_{0}\). This means that there exists a positive time \(T^{*} \in (0,T_{0}]\) such that
$$\begin{aligned} \lim_{t \to T^{*}} E\bigl[F(t)\bigr] = + \infty . \end{aligned}$$
As for the case when \(P(\tau_{\infty }=+\infty ) < 1\) (i.e., \(P(\tau_{\infty } < + \infty )>0\)), then \(\Vert \nabla u(t) \Vert \) blows up in finite time \(T^{*} \in (0, \tau_{\infty })\) with positive probability. Thus, the proof of Theorem 3.1 is completed. □

4 Global existence

In this section, we show that the solution of (1.1) is global if \(q \geq p\). We use the Borel-Cantelli lemma to prove the existence of a global solution. For this goal, we introduce an energy function
$$\begin{aligned} e\bigl(u(t)\bigr)= \bigl\Vert u_{t} (t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert \nabla u(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla u_{t} (t) \bigr\Vert ^{2} + \bigl\Vert u(t) \bigr\Vert _{p}^{p} +(h \circ \nabla u) (t). \end{aligned}$$
(4.1)

Theorem 4.1

Assume that \((u_{0},u_{1})\in H_{0}^{1} (D) \times L^{2} (D)\), \(E\int_{0}^{T} \Vert \sigma (t) \Vert ^{2}\,dt <\infty \), and condition (2.1) holds. If \(q\geq p\), \(u(t)\) is a solution of (1.1) with the initial data \((u_{0},u_{1})\in H_{0}^{1} (D) \times L^{2} (D)\) according to Definition 2.1 on the interval \([0,T]\), then for any \(T>0\), we have
$$ E \sup_{ 0 \leq t \leq T} e\bigl(u(t)\bigr) < \infty . $$
(4.2)

Proof

For any \(T>0\), we will show that \(u_{N}(t) = u(t \wedge \tau_{N} ) \to u(t)\) (a.e.) as \(N \to \infty \) for any \(t\leq T\), so that the local solution becomes a global solution where \(\tau_{N}\) is a stopping time which is defined in Section 3. Similarly to Theorem 12 of [23], we can derive the proof of the theorem. □

Declarations

Availability of data and materials

No availability.

Funding

This work was supported by the National Research Foundation of Korea (Grant ♯ NRF-2016R1D1A1B03930361). This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT and Future Planning (No. 2017R1E1A1A03070224).

Authors’ contributions

The article is a joint work of the three authors, who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Ethics approval and consent to participate

All parts in this manuscript were approved by the authors.

Competing interests

The authors declare to have no competing interests.

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This work was consent for publication.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Hankuk University of Foreign Studies, Yongin, South Korea
(2)
Department of Mathematics, Pusan National University, Busan, South Korea
(3)
Institute of Liberal Education, Catholic University of Daegu, Gyeongsan, South Korea

References

  1. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) View ArticleMATHGoogle Scholar
  2. Lions, JL: Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires. Dunod, Paris (1969) MATHGoogle Scholar
  3. Liu, W: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73, 1890-1904 (2010) MathSciNetView ArticleMATHGoogle Scholar
  4. Song, H: Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal. 125, 260-269 (2015) MathSciNetView ArticleMATHGoogle Scholar
  5. Hao, J, Wei, H: Blow-up and global existence for solution of quasilinear viscoelastic wave equation with strong damping and source term. Bound. Value Probl. 2017, 65 (2017) MathSciNetView ArticleMATHGoogle Scholar
  6. Chow, PL: Stochastic wave equations with polynomial nonlinearity. Ann. Appl. Probab. 12, 361-381 (2002) MathSciNetView ArticleMATHGoogle Scholar
  7. Chow, PL: Nonlinear stochastic wave equations: blow-up of second moments in \(L_{2}\)-norm. Ann. Appl. Probab. 19, 2039-2046 (2009) MathSciNetView ArticleMATHGoogle Scholar
  8. Bo, LJ, Tang, D, Wang, YG: Explosive solutions of stochastic wave equations with damping on \(R^{d}\). J. Differ. Equ. 244, 170-187 (2008) View ArticleMATHGoogle Scholar
  9. Chow, PL: Asymptotics of solutions to semilinear stochastic wave equations. Ann. Appl. Probab. 16, 757-789 (2006) MathSciNetView ArticleMATHGoogle Scholar
  10. Chow, PL: Asymptotic solutions of a nonlinear stochastic beam equation. Discrete Contin. Dyn. Syst., Ser. B 6, 735-749 (2006) MathSciNetView ArticleMATHGoogle Scholar
  11. Brzeźniak, Z, Maslowski, B, Seidler, J: Stochastic nonlinear beam equations. Probab. Theory Relat. Fields 132, 119-149 (2005) MathSciNetView ArticleMATHGoogle Scholar
  12. Carmona, R, Nualart, D: Random non-linear wave equation: smoothness of solutions. Probab. Theory Relat. Fields 95, 87-102 (1993) View ArticleGoogle Scholar
  13. Crauel, H, Debussche, A, Flandoli, F: Random attractors. J. Dyn. Differ. Equ. 9, 307-341 (1997) MathSciNetView ArticleMATHGoogle Scholar
  14. Dalang, R, Frangos, N: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(I), 187-212 (1998) MathSciNetMATHGoogle Scholar
  15. Duncan, T, Pasik-Duncan, B: A direct approach to linear-quadratic stochastic control. Opusc. Math. 37, 821-827 (2017) MathSciNetView ArticleGoogle Scholar
  16. Just, A, Stempień, Z: Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation. Opusc. Math. 36, 239-252 (2016) MathSciNetView ArticleMATHGoogle Scholar
  17. Millet, A, Morien, PL: On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution. Ann. Appl. Probab. 11, 922-951 (2001) MathSciNetView ArticleMATHGoogle Scholar
  18. Pardoux, E: Equations aux derivées partielles stochastiques nonlinéaries monotones. Thèse, Université Paris XI (1975) Google Scholar
  19. Parshad, RD, Beauregard, MA, Kasimov, A, Said-Houari, B: Global existence and finite blow-up in a class of stochastic nonlinear wave equations. Commun. Stoch. Anal. 8(3), 381-411 (2014) MathSciNetGoogle Scholar
  20. Kim, JU: On the stochastic wave equation with nonlinear damping. Appl. Math. Optim. 58, 29-67 (2008) MathSciNetView ArticleMATHGoogle Scholar
  21. Barbu, V, Prato, GD, Tubaro, L: Stochastic wave equations with dissipative damping. Stoch. Process. Appl. 117, 1001-1013 (2007) MathSciNetView ArticleMATHGoogle Scholar
  22. Da Prato, G, Zabczyk, J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) View ArticleMATHGoogle Scholar
  23. Cheng, S, Guo, Y, Tang, Y: Stochastic viscoelastic wave equations with nonlinear damping and source terms. J. Appl. Math. 2014, Article ID 450289 (2014) MathSciNetGoogle Scholar
  24. Gao, H, Guo, B, Liang, F: Stochastic wave equations with nonlinear damping and source terms. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(2), Article ID 1350013 (2013) MathSciNetView ArticleMATHGoogle Scholar
  25. Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525-546 (2001) MathSciNetView ArticleMATHGoogle Scholar
  26. Wei, T, Jiang, Y: Stochastic wave equations with memory. Chin. Ann. Math., Ser. B 31(3), 329-342 (2010) MathSciNetView ArticleMATHGoogle Scholar

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