# Stochastic Viscoelastic Wave Equations with Nonlinear Damping and Source Terms

Authors: Shuilin Cheng; Yantao Guo; Yanbin Tang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The goal of this paper is to study an initial boundary value problem of stochastic
viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions
on the initial data: the relaxation function, the indices of nonlinear damping, and
source terms and the random force, we prove the local existence and uniqueness of solution
by the Galerkin approximation method. Then, considering the relationship between the
indices of nonlinear damping and nonlinear source, we give the necessary conditions of
global existence and explosion in finite time in some sense of solutions, respectively.


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## Body

## 1. Introduction

We consider a stochastic viscoelastic wave equation with nonlinear damping and source terms(1)utt-Δu+∫0t‍h(t-τ)Δu(τ)dτ+|ut|q-2ut=|u|p-2u+εσ(x,t)∂tW(t,x),u(x,t)=0,(x,t)∈∂D×[0,T],u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
where D is a bounded domain in Rn with smooth boundary ∂D, q≥2, p≥2, ε is a given positive constant which measures the strength of noise, and W(t,x) is an infinite dimensional Wiener process, σ(x,t,ω) is L2(D)-valued progressively measurable and h is a positive relaxation function satisfying some conditions to be specified later. By simplicity, we have set equal to 1 all the coefficients in the equation different from the random force.For the deterministic case on viscoelastic wave equation, many authors studied the following problem:(2)utt-Δu+a|ut|q-2ut=b|u|p-2u,(x,t)∈D×(0,T),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
where a,b≥0. If a>0, b=0. Haraux and Zuazua [1] and Kopáčková [2] proved that the damping term assures existence of global solution and decay of solution for arbitrary initial data. If a=0, b>0, Ball [3] and Kalantarov and Ladyzhenskaya [4] gave that the source term causes finite time blow-up with the large initial data. If a>0, b>0, the interaction between damping term and source term occurs; Levine et al. [5, 6] studied the linear damping (i.e., q=2) and proved that the solution with negative initial energy blows up in finite time; Georgiev and Todorova [7] considered nonlinear damping and source terms; they showed that the solution blows up in finite time if p>q>2 for sufficiently large initial data and exists globally if q≥p>2 with large initial data. Alves et al. [8] and Rammaha [9] focused the nonlinear wave equations or systems on the influence between damping and source and described the existence, uniform decay rates, and blow-up to the solutions.In fact, lots of investigators have paid attention to the viscoelastic wave equation, which has its origin in the mathematical description of viscoelastic materials. The dynamic properties of viscoelastic materials are of great importance as they appear in many applications to natural sciences. The general viscoelastic wave equation has the following form:(3)utt-Δu+∫0t‍h(t-τ)Δu(τ)dτ+g(ut)=f(u),hhhhhhhhhhhhhhhhhhhhhi(x,t)∈D×(0,T),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
where h is a relaxation function and f, g are given functions. If f is nonlinear, g is linear; Kafini and Messaoudi [10] established a blow-up result; if f(u)=|u|pu, g(ut)=|ut|qut. Messaoudi [11] showed that the solution with negative initial energy blows up in finite time if p>q and exists globally if q≥p under suitable conditions on relaxation function h. This blow-up result has been pushed to the case of positive initial energy by Messaoudi [12]. For g(w)=-Δw, Song and Zhong [13] obtained that the solution with positive initial energy blows up in finite time. Ikehata [14] gave some remarks on the wave equations with nonlinear damping and source terms. Aassila et al. [15], Cavalcanti et al. [16], and Cavalcanti et al. [17] studied the boundary damping and proved the existence and uniform decay of the solutions. Cavalcanti et al. [18] discussed the asymptotic stability of the wave equation on a compact Riemannian manifold; they proved that the solutions of the corresponding partial viscoelastic model decay exponentially to zero under some conditions.Under the consideration of random environment, some authors investigated the following stochastic wave equation with nonlinear damping and source terms:(4)utt-Δu+g(ut)=f(u)+εσ(u,∇u,x,t)∂tW(t,x),hhhhhhhhhhhhhhhhhhhhhhhhi(x,t)∈D×(0,T),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
where W(t,x) is a V-valued R-Wiener process on some completed probability space, and R is a nonnegative operator with finite trace on V (see [19–26]). If f(w)=|w|p-2w, g(w)=-Δw or w; Bo et al. [27] showed that the solution blows up with positive probability or it is explosive in L2 sense. If g(w)=|w|q-2w, Gao et al. [28] showed that the global solution exists for q>p, and the solution blows up with positive probability or is explosive in energy sense for p>q.Recently, Wei and Jiang [29] and Liang and Gao [30] considered the following nonlinear stochastic viscoelastic wave equation with linear damping:
(5)utt-Δu+∫0t‍h(t-τ)Δu(τ)dτ+ut=|u|p-2u+εσ(u,∇u,x,t)∂tW(t,x),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
and they used the fixed point theorem to prove the existence and uniqueness of local mild solution; then by an appropriate energy inequality and estimations, they obtained the global existence and the decay estimate of the energy function of the solution and showed that the solution blows up with positive probability or it is explosive in L2 sense under some conditions.As we know, no one considers the stochastic viscoelastic wave equation (1) with the interaction between nonlinear damping and nonlinear source terms. In this paper, we study the global existence and the explosive phenomena under some suitable conditions on the nonlinear damping and nonlinear source terms.In contrast with the model in [27], we add a viscoelastic term ∫0th(t-τ)Δu(τ)dτ and use the nonlinear damping term |ut|q-2ut instead of the linear damping kut and the strong damping Δut. To the model in [28], we add a viscoelastic term ∫0th(t-τ)Δu(τ)dτ. To the model in [29, 30], we use the nonlinear damping term |ut|q-2ut instead of the linear damping ut. To the model in [12], we add a random force. In this paper, we generalize the blow-up and global existence results to the solution of (1) with interaction among viscoelastic memory, nonlinear damping, nonlinear source, and random force.This paper is organized as follows. In the next section, we recall some preliminaries related to assumptions and definitions for the solutions of the stochastic equations. In Section3, we use the Galerkin approximation method to get the local solution of stochastic viscoelastic wave equations with nonlinear damping and source terms. In Section 4, by the energy function and some estimates, we prove that the solution blows up with positive probability or it is explosive in energy sense for p>q. In the last section, we obtain the existence of global solution by the Borel-Cantelli Lemma.


## 2. Preliminaries

Let(X,∥·∥X) be a separable Hilbert space with Borel σ-algebra ℬ(X), and let (Ω,ℱ,P) be a probability space. We set H=L2(D) with the inner product and norm denoted by (·,·) and ∥·∥2, respectively. Denote by ∥·∥q the Lq(D) norm for 1≤q≤∞ and by ∥∇·∥2 the Dirichlet norm in V=H01(D) which is equivalent to H1(D) norm. We also assume that q, p satisfy
(6)q≥2,p>2,max⁡⁡{p,q}≤2(n-1)n-2,ifn≥3,q≥2,p>2,ifn=1,2.Lemma 1 (see [27]).
For allu,v∈H1(Rn) and 0<ρ≤(2/(n-2))(n≥3) or ρ>0(n=1,2), there exists a constant c1=c1(n,ρ)>0 such that
(7)∥u∥L2(ρ+1)≤c1∥u∥H1,∥uρv∥L2≤c1ρ+1∥u∥H1ρ∥v∥H1.One assumes thath:R+→R+ is a bounded nonincreasing C1 function satisfying h(0)>0, 1-∫0∞‍h(s)ds=l>0, and there exist positive constants ξ1 and ξ2 such that
(8)-ξ1h(t)≤h′(t)≤-ξ2h(t),t≥0.In this paper,E(·) stands for expectation with respect to probability measure P, W(t,x)(t≥0) is a V-valued R-Wiener process on the probability space with the covariance operator R satisfying Tr⁡(R)<∞. A complete orthonormal system {ek}k=1∞ in V with C0∶=sup⁡k≥1∥ek∥∞<∞, and a bounded sequence of nonnegative real numbers {λk}k=1∞ satisfies that Rek=λkek, k=1,2,….To simplify the computations, we assume that the covariance operatorR and Laplacian -Δ with homogeneous Dirichlet boundary condition have a common set of eigenfunctions, that is,
(9)-Δek=αkek,x∈D,ek=0,x∈∂D,
and then, for any t∈[0,T], W(t,x) has an expansion
(10)W(t,x)=∑k=1∞‍λkβk(t)ek(x),
where {βk(t)}k=1∞ are real valued Brownian motions mutually independent on (Ω,ℱ,P). Let ℋ be the set of L20=L2(R1/2V,V)-valued processes with the norm
(11)∥Φ(t)∥ℋ=(E∫0t‍∥Φ(s)∥L202ds)1/2=(E∫0t‍Tr⁡(Φ(s)RΦ*(s))ds)1/2<∞,
where Φ*(s) denotes the adjoint operator of Φ(s). For any process Φ(t)∈ℋ, we can define the stochastic integral with respect to the R-Wiener process as ∫0t‍Φ(s)dW(s), which is a martingale. For more details about the infinite dimension Wiener process and the stochastic integral, we refer the readers to [21].Definition 2.
Assume that(u0,u1)∈H01(D)×L2(D), and E∫0T‍∥σ(t)∥22dt<∞; u is said to be a solution of (1) on the interval [0,T], if (u,ut) is H01(D)×L2(D)-valued progressively measurable, (u,ut)∈L2(Ω;C([0,T];H01(D)×L2(D))), ut∈Lq((0,T)×D), and such that (1) holds in the sense of distributions over (0,T)×D for almost all ω.


## 3. Local Existence and Uniqueness

In this section, we establish the local existence and uniqueness of solution to problem (1) by the Galerkin approximation method. Set f(u)=|u|p-2u, g(s)=|s|q-2s. For each N≥1, we define a cut-off function χN∈C0∞, such that 0≤χN(s)≤1, ∥χN′(s)∥∞≤2 for s∈R, and
(12)χN(s)={1,|s|≤N,0,|s|≥N+1.DenotefN(u)=χN(∥∇u∥2)f(u) for u∈H01(D), then, Lemma 1 implies that
(13)∥fN(u)-fN(v)∥2≤CN∥u-v∥2,foru,v∈H01(D),
where CN is a constant depending only on N.For anyλ>0, the Yosida approximation of mapping g is
(14)gλ(x)=1λ(x-(I+λg)-1(x))=g(I+λg)-1(x),x∈R,
and it has the following properties (see [28, 31, 32]):
(15)gλ(x)∈C1(R),0≤gλ′(x)≤1λ,|gλ(x)|≤|g(x)|,|gλ(x)|≤1λ|x|,∀x∈R.Lemma 3 (see [28]).
Let{λn} be a sequence of positive numbers, and let {xn} be a sequence of real numbers such that λn→0 and xn→x as n→∞, then
(16)lim⁡n→∞gλn(xn)=g(x).Lemma 4 (see [33]).
LetD be a bounded domain in Rn(n≥1). Suppose that {ϕk} is a bounded sequence in Lq(D)(1<q<∞), such that ϕk(x)→ϕ(x) for almost all x∈D, for some ϕ(x)∈Lq(D). Then ϕk(x)→ϕ(x) weakly in Lq(D).Fixλ>0 and N>0; we consider the regularized initial boundary value problem
(17)utt-Δu+∫0th(t-τ)Δu(τ)dτ+gλ(ut)=fN(u)+εσ(x,t)∂tW(t,x),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-,
with the initial data
(18)(u0,u1)∈(H2(D)∩H01(D))×H01(D),
and σ(x,t) is H01(D)∩L∞(D)-valued progressively measurable such that
(19)E∫0T‍(∥∇σ(t)∥22+∥σ(t)∥∞2)dt<∞.For notational convenience, we omitD in the Hilbert space.Lemma 5.
Assume (18), (19), and the conditions on h hold. Then there is a pathwise unique solution u of (17) such that u∈L2(Ω;L∞(0,T;H2∩H01))∩L2(Ω;C([0,T];H01)), and ut∈L2(Ω  ;  L∞(0,T;H01))∩L2(Ω;C([0,T];L2)). Moreover, it holds that
(20)E(∫0T‍∫D‍∥ut∥L∞(0,T;H01)2+∥u∥L∞(0,T;H2∩H01)2hhhh+∫0T‍∫D‍gλ(ut)utdxdt)≤C,
where C denotes a positive constant independent of λ.Proof.
Letum(x,t)=∑j=1m‍am,jej(x) and let am,j(j=1,2,…,m) be the solution of the following system:
(21)(um′′(t),ej(x))+(∇um,ej)-∫0t‍h(t-τ)(∇um(τ),∇ej)dτ+(gλ(um′),ej)=(fN(um),ej)+ε(ej,σdWt),(x,t)∈D×(0,T),um(0)=∑j=1m‍(u0,ej),um′(0)=∑j=1m‍(u1,ej),x∈D.
By Ito^ formula, we have
(22)∥um′(t)∥22+∥∇um(t)∥22≤∥um′(0)∥22+∥∇um(0)∥22-2∫0t‍∫D‍∫0s‍h(s-τ)(Δum(τ))dτum′(s)dxds+2∫0t‍∫D‍fN(um)um′(s)dxds+2ε∫0t‍(um′(s),σdWs)-2∫0t‍∫D‍gλ(um′)um′(s)dxds+Tr⁡(R)C02∑j=1∞‍∫0t‍|(ej,εσ)|2ds,∥∇um′(t)∥22+∥Δum(t)∥22≤∥∇um′(0)∥22+∥Δum(0)∥22-2∫0t‍∫D‍∫0s‍h(s-τ)Δum(τ)dτΔum′(s)dxds-2∫0t‍∫D‍fN(um)Δum′(s)dxds+2ε∫0t‍(∇um′(s),∇(σdWs))+2∫0t‍∫D‍gλ(um′)Δum′(s)dxds+Tr⁡(R)C02∑j=1∞‍∫0t‍|(ej,ε∇σ)|2ds+∑j=1∞‍∑i=1∞‍λi∫0t‍|(ej,εσ∇ei)|2ds,
for all t∈[0,T] and almost all ω, where
(23)Tr⁡(R)=∑i=1∞‍λi<∞,C0=sup⁡k≥1∥ek∥∞<∞.
Using Hölder’s inequality, Young’s inequality, Poincaré’s inequality, and Lemma 1, we have
(24)-2∫D‍fN(um)um′(s)dx≤2∥fN(um)∥2∥um′∥2≤2CN∥um∥2∥um′∥2≤C∥∇um∥2∥um′∥2≤C∥∇um∥22+C∥um′∥22,
and so we have
(25)-2∫0t‍∫D‍fN(um)um′(s)dxds≤C∫0t‍∥∇um∥22ds+C∫0t‍∥um′∥22ds.
Since(26)-2∫D‍fN(um)Δum′(s)dx=2(p-1)∫D‍χN(∥∇um∥2)|um|p-2∇um∇um′dx≤2(p-1)∥∇um′∥2∥χN(∥∇um∥2)|um|p-2∇um∥2≤C(p-1)∥∇um′∥2∥∇um∥H1χN(∥∇um∥2)∥um∥H1p-2≤C(p-1)∥∇um′∥2∥Δum∥2χN(∥∇um∥2)∥∇um∥2p-2≤C(p-1)(N+1)p-2∥∇um′∥2∥Δum∥2≤C∥Δum∥22+C∥∇um′∥22,
thus we have
(27)-2∫0t‍∫D‍fN(um)Δum′(s)dxds≤C∫0t‍∥Δum∥22ds+C∫0t‍∥∇um′∥22ds,
where C depends on the fixed number N.
Due to integration by parts, we have(28)∫Dgλ(um′)Δum′dx=-∫D‍gλ′(um′)|∇um′|2dx<0.
Since gλ(x) is the Yosida approximation of mapping g(x), by the Lemma 3 and sign-preserving theorem of limit, for small enough λ, we have
(29)(gλ(um′(t)),um′(t))≥0.
Moreover, by the conditions of h, we get
(30)∫0t‍∫D‍h(t-τ)Δum(τ)um′(t)dxdτ=-ddt∫0t‍h(t-τ)(∇um(τ),∇um(t))dτ+h(0)(∇um(t),∇um(t))+∫0t‍h′(t-τ)(∇um(τ),∇um(t))dτ,(31)-2∫0t‍∫0s‍(h(s-τ)Δum(τ),um′(s))dτds=2∫0t‍h(t-τ)(∇um(τ),∇um(t))dτ-2h(0)∫0t‍(∇um(s),∇um(s))ds-2∫0t‍∫0s‍h′(s-τ)(∇um(τ),∇um(s))dτds,2∫0t‍∫0s‍(h(s-τ)Δum(τ),Δum′(s))dτds=2∫0t‍h(t-τ)(Δum(τ),Δum(t))dτ-2h(0)∫0t‍(Δum(s),Δum(s))ds-2∫0t‍∫0s‍h′(s-τ)(Δum(τ),Δum(s))dτds.
Next, from the properties of h, we have
(32)2∫0t‍h(t-τ)(∇um(τ),∇um(t))dτ≤12∥∇um(t)∥22+C∥∫0th(t-τ)∇um(τ)dτ∥22≤12∥∇um(t)∥22+C∫0t‍∥∇um(τ)∥22dτ,(33)-2h(0)∫0t‍(∇um(s),∇um(s))ds=-2h(0)∫0t‍∥∇um(s)∥22ds≤0,(34)|-2∫0t‍∫0s‍h′(s-τ)(∇um(τ),∇um(s))dτds|≤2∫0t‍∫0s‍|h′(s-τ)|∥∇um(τ)∥2∥∇um(s)∥2dτds≤C∫0t‍∫0s‍h(s-τ)∥∇um(τ)∥2∥∇um(s)∥2dτds≤Ch(0)∫0t‍∫0s‍∥∇um(τ)∥2dτ∥∇um(s)∥2ds≤Ch(0)(∫0t‍∥∇um(τ)∥2dτ)2≤Ch(0)T∫0t‍∥∇um(τ)∥22dτ≤C∫0t‍∥∇um(τ)∥22dτ,(35)|-2∫0t‍h(t-τ)(Δum(τ),Δum(t))dτ|≤12∥Δum(t)∥22+C∫0t‍∥Δum(τ)∥22dτ,
and similar to the derivation of (34), we have
(36)2∫0t‍∫0s‍h′(s-τ)(Δum(τ),Δum(s))dτds≤C∫0t‍∥Δum(τ)∥22dτ.
By the B-D-G inequality and Young’s inequality, we have
(37)E(sup⁡t∈[0,T]|∫0t‍(um′(s),εσdWs)|)hhhh≤εCE([∑i=1∞‍∫0T‍(σ(x,t)Rei,σ(x,t)ei)dt]1/2sup⁡t∈[0,T]∥um′(t)∥2hhhhhhhhhhhh×[∑i=1∞‍∫0T‍(σ(x,t)Rei,σ(x,t)ei)dt]1/2)hhhh≤12Esup⁡t∈[0,T]∥um′(t)∥22+ε2CE∫0T‍∥σ(t)∥22dt,E(sup⁡t∈[0,T]|∫0t‍(∇um′(s),ε∇(σdWs))|)≤12Esup⁡t∈[0,T]∥∇um′(t)∥22+ε2CE∫0T‍(∥∇σ(t)∥22+∥σ(t)∥∞2)dt.
From (22)–(37), we get
(38)Esup⁡t∈[0,T](∥um′(t)∥22+∥∇um(t)∥22hhhhhhhh+∥∇um′(t)∥22+∥Δum(t)∥22)hhhh≤∥um′(0)∥22+∥∇um(0)∥22hhhhhh+∥∇um′(0)∥22+∥Δum(0)∥22hhhhhh-2E∫0T‍(gλ(um′(s)),um′(s))dshhhhhh+12Esup⁡t∈[0,T]∥∇um∥22+12Esup⁡t∈[0,T]∥um′∥22hhhhhh+12Esup⁡t∈[0,T]∥Δum∥22+12Esup⁡t∈[0,T]∥∇um′∥22hhhhhh+3CE∫0T‍(∥um′(s)∥22+∥∇um(s)∥22hhhhhhhhhhhhhhhh+∥∇um′(s)∥22+∥Δum(s)∥22)dshhhhhh+ε2CE∫0T‍(∥σ(t)∥22+∥∇σ(t)∥22+∥σ(t)∥∞2)dthhhhhh+Tr⁡(R)C02∑j=1∞‍∫0t‍|(ej,εσ)|2dshhhhhh+Tr⁡(R)C02∑j=1∞‍∫0t‍|(ej,ε∇σ)|2dshhhhhh+∑j=1∞‍∑i=1∞‍λi∫0t‍|(ej,εσ∇ei)|2ds,
then,
(39)Esup⁡t∈[0,T](∥um′(t)∥22+∥∇um(t)∥22hhhhhhhi+∥∇um′(t)∥22+∥Δum(t)∥22)hhhh≤2(∥um′(0)∥22+∥∇um(0)∥22hhhhhhhhh+∥∇um′(0)∥22+∥Δum(0)∥22)hhhhhh-4E∫0T‍(gλ(um′(s)),um′(s))dshhhhhh+6CE∫0T‍(∥um′(s)∥22+∥∇um(s)∥22hhhhhhhhhhhhhhhhi+∥∇um′(s)∥22+∥Δum(s)∥22)dshhhhhh+C(T,∥σ∥L2([0,T]×Ω;H01),∥σ∥L2([0,T]×Ω;L∞)).
Set
(40)y(t)=∥um′(t)∥22+∥∇um(t)∥22+∥∇um′(t)∥22+∥Δum(t)∥22,
and we can rewrite (39) as follows:
(41)Esup⁡t∈[0,T]y(t)≤C∫0T‍Esup⁡t∈[0,s]y(t)ds-4E∫0T‍(gλ(um′),um′)ds+C,
and due to Gronwall’s inequality and (29), it is clear that
(42)E(sup⁡t∈[0,T](∥um′∥22+∥∇um∥22+∥∇um′∥22+∥Δum∥22)+∫0T‍(gλ(um′),um′)ds)≤C.
LetPm be the orthogonal projection of L2(D) into the space Span {e1,…,em}, such that
(43)Pmψ=∑i=1m‍(ψ,ei)ei.
Define M(t)∶=∫0t‍σdWs, where σ(x,t) is an H01(D)∩L∞(D)-valued progressively measurable such that (19) holds and {W(t,x):t≥0} is an H01-valued process; there is a subset Ω1∈Ω with P(Ω1)=1 such that for each ω⊆Ω1, M∈C([0,T];H01), and we have
(44)∂t(um′-εPmM(t))=-∫0t‍h(t-τ)Δum(τ)dτ+Δum-Pmgλ(um′)+PmfN(um)
for all m≥1.
From (42), there is a subsequence {umk}k=1∞, for each ω∈Ω1, such that,
(45)umk⇀u,weaklystarinL∞(0,T;H2∩H01),umk⟶u,stronglyinC(0,T;H01),umk′⇀u′,weaklystarinL∞(0,T;H01),hhhhhhhhhhhhhhhhhhhiiforu=u(ω),
and by the properties of relaxation function h and Hölder’s inequality, we get
(46)∥∫0t‍h(t-τ)Δum(τ)dτ∥Lq/(q-1)(0,T;H-1)q/(q-1)≤C∫0T‍(∫0s‍h(τ)dτ)1/(q-1)×∫0s‍h(s-τ)∥Δum(τ)∥H-1q/(q-1)dτds≤Csup⁡t∈[0,T]∥Δum(t)∥2q/(q-1)∫0T‍(∫0s‍h(τ)dτ)q/(q-1)ds≤C.
From (15) and embedding theorem we have
(47)∥gλ(umk′(t))∥Lq/(q-1)(0,T;H-1)q/(q-1)≤C.
Together with (42)–(47), we obtain
(48)∥umk′(t)-εPmkM(t)∥W1,q/(q-1)(0,T;H-1)≤C,
for all k≥1. By (48) and umk′(t)⇀u′(t) weakly star in L∞(0,T;H01), we have
(49)umk′(t)-εPmkM(t)⟶u′-εM(t),hhhhhhhiistronglyinC([0,T];L2).
This implies that there exists a subsequence still denoted by {umk′(t)} such that
(50)umk′(t)⟶u′(t),foralmostall(t,x)∈(0,T)×D.
Due to (47) and Lemma 4, it is clear that
(51)hhgλ(umk′(t))⟶gλ(u′(t)),weaklyinLq/(q-1)((0,T)×D).
Thus, u=u(ω) satisfies (17) in the sense of distributions over (0,T)×D.
Next, we will prove the uniqueness of the solution. If there is another solutionu~(ω) of (17), ω∈Ω1, in the above sense, then w=u-u~ satisfies
(52)wtt-Δw+∫0t‍h(t-τ)Δw(τ)dτ+gλ(ut)-gλ(ut~)=fN(u)-fN(u~),w(0)=0,wt(0)=0.
Taking the inner product of (51) with w′(t) in L2(D), we obtain
(53)12ddt(∥w′(t)∥22+∥∇w(t)∥22)hhhhh+∫0t‍h(t-τ)(Δw(τ),w′(t))dτhhhh+(gλ(ut)-gλ(ut~),w′(t))hh=(fN(u)-fN(u~),w′(t)).
From (15), we have
(54)(gλ(ut)-gλ(ut~),w′(t))≥0.
By Lemma 1 and Ho¨lder’s inequality,
(55)|(fN(u)-fN(u~),w′(t))|≤∥fN(u)-fN(u~)∥2∥w′(t)∥2≤CN∥∇w∥2∥w′(t)∥2.
Due to (30), we have
(56)∫0t‍h(t-τ)(Δw(τ),w′(t))dτ=-ddt∫0t‍h(t-τ)(∇w(τ),∇w(t))dτ+h(0)∥∇w(t)∥22+∫0t‍h′(t-τ)(∇w(τ),∇w(t))dτ.
Combining (52) with (55), similar to (32) and (34), we get
(57)∥w′∥22+∥∇w∥22≤CN∫0t‍(∥∇w∥22+∥w′∥22)ds+∫0t‍h(t-τ)(∇w(τ),∇w(t))dτ+∫0t‍∫0s‍h′(s-τ)(∇w(τ),∇w(s))dτds≤CN∫0t‍(∥∇w∥22+∥w′∥22)ds+ε∥∇w∥22+Cε∫0t‍∥∇w∥22ds
which implies w=0, that is, u(ω)=u~(ω). So u=u(ω) is well defined, for each ω∈Ω1.
Finally, we state that(u,ut) is (H2(D)∩H01(D))×H01(D)-valued progressively measurable for any 0≤t≤T, and the energy inequality holds true; this can be established by the similar argument in [28, 31].Moreover, we still fixN>0 and consider the following problem:
(58)utt-Δu+∫0t‍h(t-τ)Δu(τ)dτ+g(ut)=fN(u)+εσ(x,t)∂tW(t,x),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈D-.
The following lemma is important to prove the local existence of solution of (1).Lemma 6 (see [28, 31]).
Assume that (18), (19), and the conditions of h hold. Then there is a pathwise unique solution u of (58) such that
(59)u∈L2(Ω;L∞(0,T;H2∩H01))∩L2(Ω;C([0,T];H01)),ut∈L2(Ω;L∞(0,T;H01))∩L2(Ω;C([0,T];L2)),hhhhhhhhhhhhhhhhhhhhhhiut∈Lq((0,T)×D).
From Lemmas 5 and 6, we state a local existence theorem of (1); the proof is standard; for more information we refer the readers to [28, 31].Theorem 7 (see [28, 31]).
Assume that(u0,ut)∈H01(D)×L2(D), E∫0T‍∥σ(t)∥22dt<∞, (6), and the conditions of h hold; there is a pathwise unique local solution u of (1) according to Definition 2 such that the following energy equation holds:
(60)∥u′(t)∥22+∥∇u(t)∥22=∥u′(0)∥22+∥∇u(0)∥22-2∫0t‍∫D‍∫0s‍h(t-τ)Δu(τ)dτu′(s)dxds+2∫0t‍∫D‍|u|p-2uu′(s)dxds+2∫0t‍(u′(s),εσdWs)-2∫0t‍∫D‍|us|qdxds+ε2∑j=1∞‍∫0t‍∫D‍λjej2σ2dxds.


## 4. Blow Up

In this section we prove our main result forp>q. For this purpose, we give refined restrictions on σ(x,t) and relaxation function h such that
(61)E∫0∞‍∫D‍σ2(x,t)dxdt<∞,∫0∞‍h(s)ds<p(p-2)(p-1)2.
Define an energy function
(62)F(t)=12∥ut(t)∥22+12(1-∫0t‍h(s)ds)∥∇u(t)∥22+12(h∘∇u)(t)-1p∥u(t)∥pp,
where
(63)(h∘∇u)(t)=∫0t‍h(t-s)∥u(t)-u(s)∥22ds.For eachN, introduce the stopping time τN by τN=inf⁡{t>0;∥∇u∥2≥N}, where τN is increasing in N, let τ∞=lim⁡N→∞τN.In order to prove our blow-up result, we rewrite (1) as an equivalent Ito^ system
(64)du=vdt,dv=(Δu-∫0t‍h(t-τ)Δu(τ)dτ-|v|q-2v+|u|p-2u)dt+εσ(x,t)dW(t,x),u(x,t)=0,(x,t)∈∂D×(0,T),u(x,0)=u0(x),v(x,0)=v0(x)=u1(x),x∈D-,
where (u0,u1)∈H01×L2. Then the energy function F(t) becomes
(65)F(t)=12∥v(t)∥22+12(1-∫0t‍h(s)ds)∥∇u(t)∥22+12(h∘∇u)(t)-1p∥u(t)∥pp.First we give a lemma.Lemma 8.
Assume that (6), (61), and the conditions of h hold. Let (u,v) be a solution of system (64) with initial data (u0,v0)∈H01×L2. Then we have
(66)ddtE[F(t)]≤-E∥v∥qq+12ε2∑j=1∞‍E∫D‍λjej2(x)σ2(x,t)dx,(67)E[(u,v)(t)]=(u0,v0)-∫0t‍E∥∇u∥22ds-∫0t‍E(u,|v|q-2v)ds+∫0t‍E∥u∥ppds+E∫0t‍∫0s‍h(s-τ)(∇u(τ),∇u(s))dτds+∫0t‍E∥v∥22ds.Proof.
UsingIto^ formula to ∥v∥22 and (u,v), respectively, and taking the expectations, in the same way as our discussions in existence of solution to deal with the memory term, it is easy to get (66) and (67) (see [28]).Let(68)G(t)=ε22∑j=1∞‍E∫0t‍∫D‍λjej2(x)σ2(x,s)dxds.
Due to (61), we have
(69)G(∞)=ε22∑j=1∞‍E∫0∞‍∫D‍λjej2(x)σ2(x,s)dxds≤ε22Tr⁡(R)C02E∫0∞‍∫D‍σ2(x,s)dxds=E1<∞.
We set H(t)∶=G(t)-E[F(t)]. Then, (66) implies that
(70)H′(t)=G′(t)-ddtE[F(t)]≥E∥v(t)∥qq≥0.Lemma 9.
Let(u,v) be a solution of system (64). Then there exists a positive constant C>0 such that
(71)E∥u(t)∥ps≤C[12(h∘∇u)(t)G(t)-H(t)-E∥v(t)∥22hhhhhhhhhhhhi+E∥u(t)∥pp-12(h∘∇u)(t)],hhhhhhhhhhhhhhhhhhhhhhhhhhhi2≤s≤p.Proof.
If∥u∥p≤1 then ∥u∥ps≤∥u∥p2≤C∥∇u∥22 by Sobolev embedding theorem. If ∥u∥p≥1 then ∥u∥ps≤∥u∥pp. Therefore, combination with the definition of energy function, we can get (71).Theorem 10.
Assume that (6), (61), and the conditions of h hold. Let (u,v) be a solution of system (64) with initial data (u0,v0)∈H01×L2 satisfying
(72)F(0)≤-(1+β)E1,
where β>0 is an arbitrary constant. If p>q, then the solution (u,v) and the lifespan τ∞ defined above, either(1)
P(τ∞<+∞)>0, that is, ∥∇u(t)∥2 blows up in finite time with positive probability, or(2)
there exists a positive timeT*∈(0,T0] such that
(73)lim⁡t→T*E[F(t)]=+∞,
where
(74)T0=1-ααKL-α/(1-α)(0),L(0)=H1-α(0)+δE(u0,u1)>0,
and α,K are given in later.Proof.
For the lifespanτ∞ of the solution {u(t);t≥0} of (1) with H01 norm, firstly we treat the case when P(τ∞=+∞)<1. Then, for sufficiently large T>0, by (70) and (72), we have
(75)0<(1+β)E1≤-F(0)=H(0)≤H(t)≤G(t)+1p∥u∥pp≤E1+1p∥u∥pp.
DefineL(t)=H1-α(t)+δE(u,v), where
(76)0<α<min⁡{12,p-22p,p-qpq},
and δ is a very small constant determined in later.
Using (67) and (70), we obtain
(77)L′(t)=(1-α)H-α(t)H′(t)hhh+δ[∫0t‍h(t-τ)(∇u(τ),∇u(t))dτ+E∥v∥22-E∥∇u∥22-E(u,|v|q-2v)+E∥u∥pphhhhhhhh+E∫0t‍h(t-τ)(∇u(τ),∇u(t))dτ+E∥v∥22]≥(1-α)H-α(t)E∥v∥qq+δp[H(t)-G(t)+EF(t)]hhh-δE∥∇u∥22+δE∥v∥22-δE(u,|v|q-2v)+δE∥u∥pphhh+δ∫0t‍h(t-τ)(∇u(τ),∇u(t))dτ≥(1-α)H-α(t)E∥v∥qqhhh+δpH(t)+δ(p2-1)E∥∇u∥22hhh+δ(p2+1)E∥v∥22-δE(u,|v|q-2v)hhh+δE∫0t‍h(t-τ)(∇u(τ),∇u(t))dτhhh+δp2(h∘∇u)(t)hhh-δp2E∫0t‍h(τ)dτ∥∇u(t)∥22-δpG(t),(78)δE∫0t‍h(t-τ)(∇u(τ),∇u(t))dτ=δE∫0t‍h(t-τ)(∇u(τ)-∇u(t),∇u(t))dτ+δE∫0t‍h(τ)dτ∥∇u(t)∥22,
and by the Ho¨lder’s inequality,
(79)δE∫0t‍h(t-τ)(∇u(τ)-∇u(t),∇u(t))dτhhh≥-δE[p2∫0t‍h(t-τ)∥∇u(τ)-∇u(t)∥22dτhhhhhhhhhhh+12p∫0t‍h(τ)dτ∥∇u(t)∥22]hhh=-Eδp2(h∘∇u)(t)-δ2pE∫0t‍h(τ)dτ∥∇u(t)∥22.
Inserting (78) and (79) into (77), we get
(80)L′(t)≥(1-α)H-α(t)E∥v∥qq+δpH(t)+δ(p2-1)E∥∇u∥22+δ(p2+1)E∥v∥22-δpG(t)+δ(1-p2+12p)×∫0t‍h(τ)dτ∥∇u(t)∥22-δE(u,|v|q-2v).
From q<p, by E∥u∥qq≤CE∥u∥pq and Ho¨lder’s inequality, we obtain the estimate of the last term in (80)
(81)|E(u,|v|q-2v)|≤(E∥v∥qq)(q-1)/q(E∥u∥qq)1/q≤C(E∥v∥qq)(q-1)/q(E∥u∥pq)1/q≤C(E∥v∥qq)(q-1)/q(E∥u∥pp)1/p≤C(E∥v∥qq)(q-1)/q(E∥u∥pp)1/q×(E∥u∥pp)(1/p)-(1/q),
and the Young’s inequality implies that
(82)(E∥v∥qq)(q-1)/q(E∥u∥pp)1/q≤q-1qμE∥v∥qq+μ1-qqE∥u∥pp,
where μ is a constant determined in later.
In view of (75), we have
(83)E∥u∥pp≥p(H(t)-G(t))≥ρH(t),
where ρ=pβ/(1+β). We assume H(0)>1, (83), and (76) imply that
(84)(E∥u∥pp)(1/p)-(1/q)≤ρ(1/p)-(1/q)H(1/p)-(1/q)(t)≤ρ(1/p)-(1/q)H-α(t)≤ρ(1/p)-(1/q)H-α(0).
Combining (82) with (84), we arrive that
(85)|E(u,|v|q-2v)|≤a1q-1qμE∥v∥qqH-α(t)+a1μ1-qqE∥u∥ppH-α(0),
where a1=Cρ(1/p)-(1/q).
Hence, substituting (85) into (80),
(86)L′(t)≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qq+δpH(t)+δ(p2-1)E∥∇u∥22+δ(p2+1)E∥v∥22-δpG(t)+δ(1-p2+12p)×∫0t‍h(τ)dτ∥∇u(t)∥22-δa1μ1-qqE∥u∥ppH-α(0).
By Lemma 9 with s=p and (86), we have
(87)L′(t)≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qqhhh+δpH(t)+δ(p2-1)E∥∇u∥22hhh+δ(p2+1)E∥v∥22-δpG(t)hhh+δ(1-p2+12p)∫0t‍h(τ)dτ∥∇u(t)∥22hhh-δa2μ1-q[E∥u∥pp-12(h∘∇u)G(t)-H(t)-E∥v∥22hhhhhhhhhhhhh+E∥u∥pp-12(h∘∇u)]≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qqhhh+δ(p+a2μ1-q)H(t)-δ(p+a2μ1-q)G(t)hhh+δ(p2+1+a2μ1-q)E∥v∥22hhh-δa2μ1-qE∥u∥pp+δa2μ1-q2E(h∘∇u)hhh+δ[p2-1+(1-p2+12p)∫0t‍h(τ)dτ]∥∇u(t)∥22,
where a2=Ca1(H-α(0)/q).
Note that(88)H(t)≥G(t)+1p∥u∥pp-12∥v∥22-12∥∇u∥22-12(h∘∇u)(t),
and denote
(89)a3=p2-1+(1-p2+12p)∫0∞‍h(τ)dτ>0.
We write p=2a4+(p-2a4), where a4=min⁡{a1,a3}, the estimate (87) yields
(90)L′(t)≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qq+δ(p-2a4+a2μ1-q)H(t)-δ(p-2a4+a2μ1-q)G(t)+δ(p2+1-a4+a2μ1-q)E∥v∥22+δ(a2μ1-q+2a4p)E∥u∥pp+δ(a2μ1-q2-a4)E(h∘∇u)+δ(a3-a4)E∥∇u(t)∥22.
From (72) and (75), we obtain
(91)(p-2a4+a2μ1-q)G(t)≤(p-2a4+a2μ1-q)E1≤p-2a4+a2μ1-q1+βH(t).
Substituting (91) into (90), we get
(92)L′(t)≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qq+δ(p-2a4+a2μ1-q)β1+βH(t)+δ(p2+1-a4+a2μ1-q)E∥v∥22+δ(a2μ1-q+2a4p)E∥u∥pp+δ(a2μ1-q2-a4)E(h∘∇u)+δ(a3-a4)∥∇u(t)∥22.
Next, we can chooseμ large enough so that (92) becomes
(93)L′(t)≥(1-α-a1q-1qμδ)H-α(t)E∥v∥qqhhh+δγ(H(t)+E∥v∥22+E∥u∥pphhhhhhhhi-E(h∘∇u)+∥∇u(t)∥22),
where γ>0 is the minimum of the coefficients of H(t), E∥v∥22, E∥u∥pp, E(h∘∇u), and ∥∇u(t)∥22 in (93). Once μ is fixed, we pick δ small enough so that
(94)1-α-a1q-1qμδ≥0.
Therefore, (93) takes the form
(95)L′(t)≥δγ(H(t)+E∥v∥22+E∥u∥pphhhhhh-E(h∘∇u)+∥∇u(t)∥22)≥0.
Consequently we have
(96)L(t)≥L(0)=H1-α(0)+δE(u0,u1)>0,∀t≥0.
Since
(97)|E∫D‍uvdx|≤C(E∥u∥p2)1/2(E∥v∥22)1/2,
it implies that
(98)|E∫D‍uvdx|1/(1-α)≤C{(E∥u∥p2)η/2(1-α)+(E∥v∥22)ζ/2(1-α)},
for (1/η)+(1/ζ)=1.
We chooseζ=2(1-α),η=2(1-α)/(1-2α); then η/2(1-α)=1/(1-2α)≤p/2, by (76) and (98) becomes
(99)|E∫D‍uvdx|1/(1-α)≤C{E∥u∥p2/(1-2α)+E∥v∥22}.
Using Lemma 9 with s=2/(1-2α), we obtain
(100)|E∫D‍uvdx|1/(1-α)≤C(+E∥u∥pp+E(h∘∇u)H(t)+E∥∇u∥22+E∥v∥22+E∥u∥pp+E(h∘∇u))
for all t≥0.
Therefore we have(101)L1/(1-α)(t)=(H1-α(t)+δE(u,v))1/(1-α)≤21/(1-α)(H(t)+δ1/(1-α)|E∫D‍uvdx|1/(1-α))≤C(H(t)+E∥∇u∥22+E∥v∥22hhhi+E∥u∥pp+E(h∘∇u))
for all t≥0. Combining (95) and (101),
(102)L′(t)≥KL1/(1-α)(t),∀t≥0,
where K is a positive constant depending only on C and δγ; then it yields
(103)Lα/(1-α)(t)≥1-α(1-α)L-α/(1-α)(0)-αKt.
Let(104)T0=1-ααKL-α/(1-α)(0).
Then L(t)→∞ as t→T0. This means that there exists a positive time T*∈(0,T0] such that
(105)lim⁡t→T*E[F(t)]=+∞.
As for the case when P(τ∞=+∞)<1 (i.e.,P(τ∞<+∞)>0), then ||∇u(t)||2 blows up in finite time T*∈(0,τ∞] with positive probability.
The proof of Theorem10 is completed.Remark 11.
(1) In the deterministic case ofε=0, it is well known that for (u0,u1)∈H01×L2, the condition F(0)<0 and ∫0∞‍h(s)ds<p(p-2)/(p-1)2 already imply that the solution blows up in finite time (see, e.g., [11]). In the stochastic case of ε>0, to balance the influence of W(t,x) such that the local solution of (1) blows up with positive probability or is explosive in L2 sense, the initial energy and relaxation function should be satisfied that F(0)≤-(1+β)E1, and ∫0∞‍h(s)ds<p(p-2)/(p-1)2.
(2) Our results have included the case which is without viscoelastic term (i.e.,h=0 satisfied (61)).


## 5. Global Existence

In this section we show that solution of (1) is global if q≥p. We use the Borel-Cantelli Lemma to prove the existence of global solution. For this aim, we introduce an energy function
(106)e(u(t))=∥ut(t)∥22+∥∇u(t)∥22+∥u(t)∥pp+(h∘∇u)(t).Theorem 12.
Assume that (1), (u0,u1)∈H01(D)×L2(D), E∫0T‍∥σ(t)∥22dt<∞, and the conditions of h hold. If q≥p, u(t) is a solution of (1) with initial data (u0,u1)∈H01×L2 according to Definition 2 on the interval [0,T]; then for any T>0,
(107)Esup⁡0≤t≤Te(u(t))<∞.Proof.
For anyT>0, we will show that uN(t)=u(t∧τN)→u (a.s.) as N→∞ for any t≤T, so that the local solution becomes a global solution where τN is a stopping time which is defined in Section 4. Similar to [28], by the Theorem 7, for t∈[0,T∧τN), u(t)=uN(t)=u(t∧τN) is the local solution of (1), so the following energy equation holds:
(108)e(u(t∧τN))=e(u0)+(h∘∇u)(t∧τN)-2∫0t∧τN‍∫D‍∫0s‍h(t-τ)Δu(τ)dτu′(s)dxds+4∫0t∧τN‍∫D‍|u|p-2uu′(s)dxds+2∫0t∧τN‍(u′(s),εσdWs)-2∫0t∧τN‍∫D‍|ut|qdxds+ε2∑j=1∞‍∫0t∧τN‍∫D‍λjej2σ2dxds.
Using Hölder’s inequality, Young’s inequality, and embedding theorem, we have
(109)|∫D‍|u|p-2uu′(t)dx|≤∥u∥pp-1∥ut∥p≤ν∥ut∥pp+Cν∥u∥pp≤Cν∥ut∥qp+Cν∥u∥pp,
where ν>0,C is the embedding constant, and Cν is a constant depending on ν, consequently; we have
(110)e(u(t∧τN))≤e(u0)+(h∘∇u)(t∧τN)-2∫0t∧τN‍∫D‍∫0s‍h(s-τ)Δu(τ)dτu′(s)dxds+4Cν∫0t∧τN‍∥ut∥qpds+4Cν∫0t∧τN‍∥u∥ppds+2∫0t∧τN‍(u′(s),εσdWs)-2∫0t∧τN‍∥ut∥qqds+ε2Tr⁡(R)C02∫0t∧τN‍∥σ∥22ds.
Sinceq≥p, we distinguish two cases.(1)
Either∥ut∥qq>1 so we choose ν so small that -2∥ut∥qq+4Cν∥ut∥qp≤0.(2)
Or∥ut∥qq≤1; in this case, we have -2∥ut∥qq+4Cν∥ut∥qp≤4Cν.Hence, in either case, we have
(111)e(u(t∧τN))≤e(u0)+(h∘∇u)(t∧τN)-2∫0t∧τN‍∫D‍∫0s‍h(s-τ)Δu(τ)dτutdxds+4Cν(t∧τN)+2∫0t∧τN‍(u′(s),εσdWs)+4Cν∫0t∧τN‍∥u∥ppds+ε2Tr⁡(R)C02∫0t∧τN‍∥σ∥22ds.
Using the conditions of h, we obtain
(112)-∫D‍∫0s‍h(s-τ)(Δu(τ))dτutdx=∫0s‍h(s-τ)∫D‍∇u(τ)∇ut(s)dxdτ=∫0s‍h(s-τ)∫D‍(∇u(τ)-∇ut(s))∇ut(s)dxdτ+∫0s‍h(s-τ)∫D‍∇ut(s)∇ut(s)dxdτ=-12∫0s‍h(s-τ)dds∫D‍|∇u(τ)-∇ut(s)|2dxdτ+12∫0s‍h(τ)dds∫D‍|∇u(τ)|2dxdτ=12dds(∫0s‍h(τ)dτ∥∇u(s)∥22-(h∘∇u)(s))+12(h′(s)∘∇u)(s)-12h(s)∥∇u(s)∥22≤12dds(∫0s‍h(τ)dτ∥∇u(s)∥22-(h∘∇u)(s))
which implies that
(113)-2∫0t∧τN‍∫D‍∫0s‍h(s-τ)(Δu(τ))dτut(s)dxds≤∫0t∧τN‍h(τ)dτ∥∇u(t)∥22-(h∘∇u)(t∧τN).
Consequently we have(114)e(u(t∧τN))≤e(u0)+∫0t∧τN‍h(τ)dτ∥∇u(t)∥22+4Cν(t∧τN)+2∫0t∧τN‍(u′(s),εσdWs)+4Cν∫0t∧τN‍∥u∥ppds+ε2Tr⁡(R)C02∫0t∧τN‍∥σ∥22ds.
Taking the expectation of (114), we get
(115)Ee(u(t∧τN))≤e(u0)+4Cν(t∧τN)+C∫0t∧τN‍Ee(u(s))ds+ε2Tr⁡(R)C02∫0t∧τN‍E∥σ∥22ds.
The Gronwall’s inequality implies that
(116)Ee(u(t∧τN))≤(e(u0)+CT)eCT≤CT.
On the other hand, we have(117)Ee(u(t∧τN))≥E(IτN<Te(u(τN)))≥CE(IτN<T∥∇u(τN)∥22)≥CN2P(τN<T),
where I(·) denotes the indicator function. In view of (116) and (117), we get
(118)P(τN<T)≤CTN2.
The Borel-Cantelli lemma implies that P(τ∞<T)=0 for any T>0. This shows that u=lim⁡N→∞uN(t) is the global solution.



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*Source: 450289-2014-03-27.xml*