Stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms

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.

In this paper, we are concerned with the following stochastic viscoelastic wave equation:

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 [1–5], and the references therein. For example, Liu [3] considered the following quasilinear viscoelastic wave equation problem:

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

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

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 [6–25]).

Wei and Jiang [26] and Gao, Guo and Liang [24] considered the following nonlinear stochastic viscoelastic wave equation:

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

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).

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

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

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,

and then, for any \(t \in [0,T]\), \(W(x,t)\) has an expansion

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

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

and

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

Now, we define an energy function

where

For each N, stopping time \(\tau_{N}\) is given as

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

where \((u_{0},u_{1})\in H_{0}^{1}(D)\times L^{2}(D)\). Then the energy function \(F(t)\) becomes

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

and

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

Due to (3.1), we deduce

We set

Then (3.5) implies that

Lemma 3.2

Let \((u,v)\) be a solution of Eq. (3.3). Then there exists a positive constant C such that

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

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

Define

where

and δ is a very small constant to be determined later.

Using (3.6) and (3.9), we deduce

On the other hand, we have

and by Hölder’s inequality, we get

Inserting (3.17) and (3.18) into (3.16), we obtain

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]):

and Young’s inequality

where μ is a constant to be determined later. In view of (3.14), we get

where \(\widetilde{\rho } = p\beta /(1+\beta )\). With the assumption of \(H(0)>1\), (3.21), (3.22), and (3.15) imply that

Combining with (3.20), (3.21), and (3.23), we arrive at

where \(a_{1}=C\widetilde{\rho }^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}\). Hence, substituting (3.24) for (3.19), we get

Using Lemma 3.2 with \(s=p\) and (3.25), we have

where \(a_{2}=Ca_{1} H^{-\alpha }(0)/q \).

Note that

with

we write \(p=2a_{4}+(p-2a_{4})\) with \(a_{4}=\min \{ a_{1}, a_{3}\}\), then estimate (3.26) yields

From (3.8) and (3.14), we deduce

Substituting (3.30) with (3.29), we get

Next, we can choose μ large enough so that (3.31) becomes

where

Once μ is fixed, we pick δ small enough so that

Using this, (3.32) takes the form

Thus, we see that

Consequently, we get

Since

we have

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

Using Lemma 3.2 with \(s= \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}}\), we obtain

Therefore, we deduce, for all \(t\geq 0\),

Combining (3.33) and (3.37)

with a positive constant K depending on C and δγ, it follows that

Let

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

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. □

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

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

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. □

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).

Affiliations

1. Hankuk University of Foreign Studies, Yongin, South Korea

   • Sangil Kim

2. Department of Mathematics, Pusan National University, Busan, South Korea

   • Jong-Yeoul Park

3. Institute of Liberal Education, Catholic University of Daegu, Gyeongsan, South Korea

   • Yong Han Kang

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.

Corresponding author

Correspondence to Yong Han Kang.

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.

Consent for publication

This work was consent for publication.

Abbreviations

Stochastic quasilinear viscoelastic wave equation.

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