Kim et al. Boundary Value Problems (2018) 2018:14 https://doi.org/10.1186/s13661-018-0932-z

RESEARCH

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Stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms Sangil Kim1 , Jong-Yeoul Park2 and Yong Han Kang3* *

Correspondence: [email protected] 3 Institute of Liberal Education, Catholic University of Daegu, Gyeongsan, South Korea Full list of author information is available at the end of the article

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 |ut |q–2 ut and a source term of the type |u|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, ρ + 2} and the initial data are large enough (that is, if the initial energy is sufficiently negative). Also, we show that if q ≥ p, the local solution can be extended for all time and is thus global. MSC: 60H15; 35L05; 35L70 Keywords: stochastic quasilinear viscoelastic wave equation; blow-up of solution; global existence

1 Introduction In this paper, we are concerned with the following stochastic viscoelastic wave equation:  |ut |ρ utt – u – utt +

t

h(t – τ )u(τ ) dτ + |ut |q–2 ut

0

= |u|p–2 u + σ (x, t)∂t W (x, t) in D × (0, T), u=0

(1.1)

on ∂D × (0, T),

u(x, 0) = u0 (x),

¯ ut (x, 0) = u1 (x) in D,

where D is a bounded domain in Rn with smooth boundary ∂D, with given positive constants ρ > 0, q ≥ 2, and p ≥ 2. The function h : R+ → 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, σ (x, t) is L2 (D)-valued progressively measurable, and  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 © The Author(s) 2018. 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.

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wave equation problem: 

t

|ut |ρ utt – u – utt +

g(t – τ )u(τ ) dτ = b|u|p–2 u in D × (0, ∞),

0

u=0

on ∂D × (0, ∞),

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x) in ,

where D is a bounded domain in Rn (n ≥ 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 

t

|ut | utt – u + ρ

g(t – τ )u(τ ) dτ + |ut |m–2 ut = |u|p–2 u

in D × [0, T],

0

u=0

on ∂D × [0, T],

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x) in D,

where D is a bounded domain of Rn (n ≥ 1) with a smooth boundary ∂D, m > 2, g : R+ → R+ is a positive nonincreasing function, and 2 < p < ∞ if n = 1, 2,

2 < p < 2(n – 1)/(n – 2)

if n ≥ 3,

2 < ρ < ∞ if n = 1, 2,

2 < ρ ≤ n/(n – 2) if n ≥ 3.

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:  utt – u +

t

h(t – τ )u(τ ) dτ + ut 0

= |u|p–2 u + σ (u, ∇u, x, t)∂t W (x, t) in D × (0, T), u=0

on ∂D × (0, T),

u(x, 0) = u0 (x),

¯ ut (x, 0) = u1 (x) in D.

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 L2 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]).

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Recently, Cheng et al. [23] studied the stochastic viscoelastic wave equation with nonlinear damping and source terms 

t

utt – u +

h(t – τ )u(τ ) dτ + |ut |q–2 ut

0

= |u|p–2 u + σ (x, t)∂t W (x, t) in D × (0, T), u=0

on ∂D × (0, T),

u(x, 0) = u0 (x),

¯ ut (x, 0) = u1 (x) in D,

where D is a bounded domain in Rn with smooth boundary ∂D, q ≥ 2, p ≥ 2,  is a given positive constant which measures the strength of noise; W (x, t) is an infinite dimensional Wiener process; σ (x, t, w) is L2 (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, ρ + 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, ρ + 2}. In Section 4, we obtain global existence of equation (1.1).

2 Preliminaries Let (X,  · X ) be a separable Hilbert space with Borel σ -algebra B(X), and let (, F, P) be a probability space. We set H = L2 (D) with the inner product and norm denoted by (·, ·) and  · , respectively. We denote by  · q the Lq (D) norm for 0 ≤ q ≤ ∞ and by ∇ ·  the Dirichlet norm in V = H01 (D) which is equivalent to H 1 (D) norm. First, we introduce the following hypotheses: (H1) We assume that p, q, ρ satisfy q ≥ 2,

p > 2,

q ≥ 2,

p>2

0≤ρ≤

2 n–2

max{p, q} ≤

2(n – 1) n–2

if n ≥ 3; (2.1)

if n = 1, 2; if n ≥ 3,

0 < ρ < ∞ if n = 1, 2.

(H2) We assume that h : R+ → R+ is a bounded nonincreasing C 1 function satisfying  h(s) > 0,

∞

h(s) ds = l > 0,

1– 0

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and there exist positive constants ξ1 and ξ2 such that –ξ1 h(t) ≤ h (t) ≤ –ξ2 h(t),

t ≥ 0.

(2.2)

(H3) σ (x, t) is H01 (D) ∩ L∞ (D)-valued progressively measurable such that  E

    ∇σ (t)2 + σ (t)2 dt < ∞. ∞

T 

0

Lemma 2.1 ([8]) For all u, v ∈ H 1 (Rn ) and 0 < p ≤ exists a constant c1 = c1 (n, p) > 0 such that uL2(p–1) ≤ c1 uH 1 ,

(2.3) 2 n–2

(n ≥ 3) or p > 0 (n = 1, 2), there

 p  u v ≤ cp+1 up 1 vH 1 . 1 H

(2.4)

In this paper, E(·) stands for expectation with respect to probability measure P, and W (x, t)(t ≥ 0) is a V -valued Q-Wiener process on the probability space with the covariance operator Q satisfying Tr(Q) < ∞. A complete orthonormal system {ek }∞ k=1 in V with c0 := supk≥1 ek ∞ < ∞ and a bounded sequence of nonnegative real members {λk }∞ k=1 satisfy that Qek = λk ek ,

k = 1, 2, . . . .

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, –ek = μk ek , ek = 0,

x ∈ D,

x ∈ ∂D,

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

∞  

λk βk (t)ek (t),

(2.5)

k=1

where {βk (t)}∞ k=1 are real-valued Brownian motions mutually independent of (, F, P). Let H be the set of L02 = L2 (Q1/2 V , V )-valued processes with the norm   t 1/2   t 1/2       ∗  (t) = E  (s)2 0 ds = E Tr (s)Q (s) ds < ∞, L H 0

2

(2.6)

0

where ∗ (s) denotes the adjoint operator of (s). For any ∗ (t) ∈ H, we can define the

t stochastic integral with respect to the Q-Wiener process as 0 (s) dW (s), which is martingale. For more details about the finite dimension Winner process and the stochastic integral, see [22].

T Definition 2.1 Assume that (u0 , u1 ) ∈ H01 (D) × L2 (D) and E 0 σ (t)2 dt < ∞. u is said to be the solution of (1.1) on the interval [0, T] if (u, ut ) is H01 (D) × L2 (D)-valued progressively

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measurable, (u, ut ) ∈ L2 (; C([0, T]; H01 (D) × L2 (D))), ut ∈ Lq ((0, T) × D), and such that (1.1) holds in the sense of distributions over (0, T) × 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 (u0 , u1 ) ∈ (H 2 (D) ∩ H01 (D)) × H01 (D), problem (1.1) has a pointwise unique solution u such that       u ∈ L2 ; L∞ 0, T; H 2 (D) ∩ H01 (D) ∩ L2 ; C [0, T]; H01 (D) and       ut ∈ L2 ; L∞ 0, T; H01 (D) ∩ L2 ; C [0, T]; L2 (D) .

3 Blow-up result In this section, we prove our main result for p > q. For this purpose, we give defined restrictions on σ (x, t) and the relaxation function h such that 



∞

σ 2 (x, t) dx dt < ∞,

E 0

∞

h(s) ds < 0

D

p(p – 2) . (p – 1)2

(3.1)

Now, we define an energy function F(t) =

   1  ut (t)ρ+2 + 1 ∇ut (t)2 ρ+2 ρ+2 2   t p  2 1 1 1 1– + h(s) ds ∇u(t) + (h ◦ ∇u)(t) – u(t)p , 2 2 p 0

(3.2)

where  (h ◦ ∇u)(t) =

t

 2 h(t – s)∇u(t) – ∇u(s) ds.

0

For each N , stopping time τN is given as  2  τN = inf t > 0 : ∇u(t) ≥ N , where τN is increasing in N , and τ∞ = limN→∞ τN . In order to prove our blow-up result, we rewrite (1.1) as an equivalent Itô’s system du = v dt,    t 1 d |v|ρ v – v = u – h(t – s)u(s) ds – |v|q–2 v + |u|p–2 u dt ρ +1 0 + σ (x, t) dWt (x, t), u(x, t) = 0,

(x, t) ∈ D × (0, T),

(x, t) ∈ ∂D × (0, T),

u(x, 0) = u0 (x),

v(x, 0) = v0 (x) = u1 (x),

x ∈ D,

(3.3)

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where (u0 , u1 ) ∈ H01 (D) × L2 (D). Then the energy function F(t) becomes F(t) =

   1  v(t)ρ+2 + 1 ∇v(t)2 ρ+2 ρ+2 2   t  2 1 1 1– + h(s) ds ∇u(t) + (h ◦ ∇u)(t) – 2 2 0

 1 u(t)p . p p

(3.4)

Lemma 3.1 Let (u, v) be a solution of Eq. (3.3) with the initial data (u0 , v0 ) ∈ H01 (D) × L2 (D). Then we have  ∞  q  2  d EF(t) = –Ev(t)q + E λj e2j (x)σ 2 (x, t) dx dt 2 j=1 D  2   1 – E –h ◦ ∇u (t) – h(t)E∇u(t) 2  ∞  q  2    ≤ –E v(t) q + E λj e2j (x)σ 2 (x, t) dx, 2 j=1 D

(3.5)

and 1



ρ v(t) v(t) – v(t) ρ+1  t   2 1 E∇u(s) ds = u0 , |u1 |ρ u1 – u1 – ρ+1 0  t  t

q–2  p   – E u(s), v(s) v(s) ds + Eu(s) ds

 E u(t),

0

 t

+E 0

+

0

1 E ρ +1

0 s

p

  h(s – τ ) ∇u(τ ), ∇u(s) dτ ds

 0

t

 v(s)ρ+2 ds + E ρ+2



t

 ∇v(s)2 ds.

(3.6)

0

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) =

 t ∞ 2  E λj e2j (x)σ 2 (x, s) dx ds. 2 j=1 D 0

(3.7)

Due to (3.1), we deduce

G(∞) =

≤

 ∞ ∞ 2  E λj e2j (x)σ 2 (x, s) dx ds 2 j=1 D 0 2 Tr(Q)c20 E 2

 0

∞ D

σ 2 (x, s) dx ds = E1 < ∞.

(3.8)

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We set   H(t) = G(t) – E F(t) . Then (3.5) implies that H  (t) = G (t) –

 q  d  E F(t) ≥ Ev(t)q ≥ 0. dt

(3.9)

Lemma 3.2 Let (u, v) be a solution of Eq. (3.3). Then there exists a positive constant C such that  ρ+2  s  Eu(t)p ≤ C G(t) – H(t) – Ev(t)ρ+2  2  p  – E∇v(t) + Eu(t) – E(h ◦ ∇u)(t) , p

2 ≤ s ≤ p.

(3.10)

Proof If up ≤ 1, then usp ≤ u2p ≤ C∇u2 by the Sobolev embedding theorem. p If up ≥ 1, then usp ≤ up . Thus there exists a constant C > 0 such that Eusp ≤ p C(E∇u2 + Eup ). 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 (u0 , v0 ) ∈ H01 (D) × L2 (D) satisfying F(0) ≤ –(1 + β)E1 ,

(3.11)

where β > 0 is an arbitrary constant and E1 is defined in (3.8). If p > {q, ρ + 2}, then the solution (u, v) and the lifespan τ∞ defined above are either (1) P(τ∞ < ∞) > 0, that is, ∇u(t) blows up in finite time with positive probability, or (2) there exists a positive time T ∗ ∈ [0, T0 ] such that   lim E F(t) = +∞,

(3.12)

t→T ∗

where T0 =

1–α , αKLα/(1–α) (0)

L(0) = H

1–α

 (0) + δE u0 ,

1 ρ |u1 | u1 – u1 > 0, ρ +1

(3.13)

and α, K are given later. Proof For the lifespan τ∞ of the solution {u(t) : t > 0} of Eq. (3.3) with H01 (D) norm, we treat the case when P(τ∞ = +∞) < 1. Then, for sufficiently large T > 0, by (3.9) and (3.11), we obtain p p 1  1  0 < (1 + β)E1 ≤ –F(0) = H(0) ≤ H(t) ≤ G(t) + Eu(t)p ≤ E1 + Eu(t)p . p p

(3.14)

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Define  L(t) = H 1–α (t) + δE u(t),

1



ρ v(t) v(t) – v(t) , ρ+1

where   1 1 p–2 p–q 1 , , – 0 < α < min , 2 2p pq ρ + 2 p

(3.15)

and δ is a very small constant to be determined later. Using (3.6) and (3.9), we deduce   2

q–2   L (t) = (1 – α)H –α (t)H  (t) + δ –E∇u(t) – E u(t), v(t) v(t)  p + Eu(t)p + E



t

  h(t – τ ) ∇u(τ ), ∇u(t) dτ

0

  2  ρ+2 1     + E v(t) ρ+2 + E ∇v(t) ρ +1  q   ≥ (1 – α)H –α (t)Ev(t)q + δp H(t) – G(t) + EF(t)

q–2  p  2   – δE∇u(t) – δE u(t), v(t) v(t) + δEu(t)p  t  ρ+2  2   δ Ev(t)ρ+2 + δE∇v(t) + δE h(t – τ ) ∇u(τ ), ∇u(t) dτ + ρ+1 0   q ≥ (1 – α)H –α (t)Ev(t)q + δpH(t)   ρ+2 1 p + Ev(t)ρ+2 +δ ρ +2 ρ+1   2 p +δ – 1 E∇u(t) 2 

q–2  2   p +δ + 1 E∇v(t) – δE u(t), v(t) v(t) 2  t   + δE h(t – τ ) ∇u(τ ), ∇u(t) dτ 0

δp δp + E(h ◦ ∇u)(t) – E 2 2



t

 2 h(τ ) dτ ∇u(t) – δpG(t).

(3.16)

0

On the other hand, we have  δE

t

  h(t – τ ) ∇u(τ ), ∇u(t) dτ

0



= δE 0

t

  h(t – τ ) ∇u(τ ) – ∇u(t), ∇u(t) dτ



+ δE 0

t

 2 h(τ ) dτ ∇u(t) ,

(3.17)

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and by Hölder’s inequality, we get  δE

t

  h(t – τ ) ∇u(τ ) – ∇u(t), ∇u(t) dτ

0

≥–

δp δ E(h ◦ ∇u)(t) – E 2 2p



t

 2 h(τ ) dτ ∇u(t) .

(3.18)

0

Inserting (3.17) and (3.18) into (3.16), we obtain  q L (t) ≥ (1 – α)H –α (t)Ev(t)q + δpH(t)   ρ+2 1 p + Ev(t)ρ+2 +δ ρ +2 ρ+1   2 p +δ – 1 E∇u(t) 2 

q–2  2   p +δ + 1 E∇v(t) – δE u(t), v(t) v(t) 2  t   2 p2 + 1 E h(τ ) dτ ∇u(t) . – δpG(t) + δ 1 – 2p 0 q

(3.19)

q

For q < p, by Eu(t)q ≤ cEu(t)p and Hölder’s inequality, we deduce the following estimate (see [23]): q  1

q–2     q  q–1   E u(t), v(t) v(t) ≤ Ev(t)q q Eu(t)q q q  1   q  q–1   ≤ C Ev(t)q q Eu(t)p q p  1   q  q–1   ≤ C Ev(t)q q Eu(t)p p p  1   p  1 – 1   q  q–1   ≤ C Ev(t)q q Eu(t)p q Eu(t)p p q

(3.20)

and Young’s inequality p  1 q – 1  q μ1–q  p   q  q–1   Ev(t)q q Eu(t)p q ≤ μEv(t)q + Eu(t)p , q q

(3.21)

where μ is a constant to be determined later. In view of (3.14), we get  p   Eu(t)p ≥ p H(t) – G(t) ≥ ρ H(t),

(3.22)

where ρ  = pβ/(1 + β). With the assumption of H(0) > 1, (3.21), (3.22), and (3.15) imply that p  1 – 1   1 1 1 1 1 1 1 1 Eu(t)p p q ≤ ρ p – q H(t) p – q ≤ ρ p – q H –α (t) ≤ ρ p – q H –α (0).

(3.23)

Combining with (3.20), (3.21), and (3.23), we arrive at 1–q 



   

E u(t), v(t) q–2 v(t) ≤ a1 q – 1 μEv(t)q H –α (t) + a1 μ Eu(t)p H –α (t), q p q q

(3.24)

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1

where a1 = C ρ p – q . Hence, substituting (3.24) for (3.19), we get   q q–1 μδ H –α (t)Ev(t)q + δpH(t) L (t) ≥ 1 – α – a1 q   ρ+2 1 p + Ev(t)ρ+2 +δ ρ +2 ρ+1   2 p – 1 E∇u(t) +δ 2   2 p +δ + 1 E∇v(t) – δpG(t) 2  t   2 p2 + 1 h(τ ) dτ E∇u(t) +δ 1– 2p 0 – δa1

p μ1–q  Eu(t)p H –α (0). q

(3.25)

Using Lemma 3.2 with s = p and (3.25), we have   q q–1 L (t) ≥ 1 – α – a1 μδ H –α (t)Ev(t)q + δpH(t) q   ρ+2 p 1 +δ + Ev(t)ρ+2 ρ +2 ρ+1   2 p – 1 E∇u(t) +δ 2   2 p + 1 E∇v(t) – δpG(t) +δ 2  t   2 p2 + 1 E h(τ ) dτ ∇u(t) +δ 1– 2p 0  ρ+2  – δa2 μ1–q G(t) – H(t) – Ev(t)ρ+2  2  p  – E∇v(t) + Eu(t)p – E(h ◦ ∇u)(t)   q q–1 μδ H –α (t)Ev(t)q ≥ 1 – α – a1 q     1–q H(t) – δ p + a2 μ1–q G(t) + δ p + a2 μ   ρ+2 p 1 1–q Ev(t)ρ+2 +δ + + a2 μ ρ +2 ρ+1   2  p p + 1 + a2 μ1–q E∇v(t) – δa2 μ1–q Eu(t)p +δ 2 

+ δa2 μ1–q E(h ◦ ∇u)(t)    t   2 p p2 + 1 +δ –1+ 1– h(τ ) dτ E∇u(t) , 2 2p 0 where a2 = Ca1 H –α (0)/q.

(3.26)

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Note that p  ρ+2 1 1  Ev(t)ρ+2 H(t) ≥ G(t) + Eu(t)p – p ρ +2 2 1  2 1 1  – E∇v(t) – ∇u(t) – E(h ◦ ∇u)(t) 2 2 2

(3.27)

with   t p2 + 1 p a3 = – 1 + 1 – h(τ ) dτ > 0, 2 2p 0

(3.28)

we write p = 2a4 + (p – 2a4 ) with a4 = min{a1 , a3 }, then estimate (3.26) yields   q q–1 L (t) ≥ 1 – α – a1 μδ H –α (t)Ev(t)q q     + δ p – 2a4 + a2 μ1–q H(t) – δ p – 2a4 + a2 μ1–q G(t)   ρ+2 p 2a4 1 +δ – + + a2 μ1–q Ev(t)ρ+2 ρ +2 ρ +2 ρ +1   2 p +δ – a4 + 1 + a2 μ1–q E∇v(t) 2  p 2a4  1–q Eu(t)p + δ –a2 μ + p  2   + δ a2 μ1–q – a4 E(h ◦ ∇u)(t) + δ(a3 – a4 )E∇u(t) . 

(3.29)

From (3.8) and (3.14), we deduce 

   p – 2a4 + a2 μ1–q G(t) ≤ p – 2a4 + a2 μ1–q E1 ≤

p – 2a4 + a2 μ1–q H(t). 1+β

(3.30)

Substituting (3.30) with (3.29), we get   q q–1 μδ H –α (t)Ev(t)q L (t) ≥ 1 – α – a1 q  β  + δ p – 2a4 + a2 μ1–q H(t) 1+β   ρ+2 p 2a4 1 +δ – + + a2 μ1–q Ev(t)ρ+2 ρ +2 ρ +2 ρ +1   2 p 1–q E∇v(t) – a4 + 1 + a2 μ +δ 2  p 2a4  Eu(t)p + δ –a2 μ1–q + p  2   + δ a2 μ1–q – a4 E(h ◦ ∇u)(t) + δ(a3 – a4 )E∇u(t) .

(3.31)

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Next, we can choose μ large enough so that (3.31) becomes   q  ρ+2  2  q–1 L (t) ≥ 1 – α – a1 μδ H –α (t)Ev(t)q + δγ H(t) + Ev(t)ρ+2 + E∇v(t) q  p  2  (3.32) + Eu(t)p + E(h ◦ ∇u)(t) + E∇u(t) , where    β p 2a4 1 γ = min p – 2a4 + a2 μ1–q , – + + a2 μ1–q , 1+β ρ +2 ρ +2 ρ +1  p 2a4 – a4 + 1 + a2 μ1–q , –a2 μ1–q + , a2 μ1–q – a4 , a3 – a4 > 0. 2 p Once μ is fixed, we pick δ small enough so that 1 – α – a1

q–1 μδ > 0. q

Using this, (3.32) takes the form  ρ+2  2  L (t) ≥ δγ H(t) + Ev(t)ρ+2 + E∇v(t)  p  2  + Eu(t)p + E(h ◦ ∇u)(t) + E∇u(t) .

(3.33)

Thus, we see that  L(t) ≥ L(0) = H 1–α (0) + δE u0 ,

1 |u1 |ρ u1 – u1 > 0, ρ +1

∀t ≥ 0.

(3.34)

Consequently, we get L(t) ≥ L(0) > 0,

∀t ≥ 0.

(3.35)

Since



   



E v(t) ρ v(t)u(t) dx ≤ Ev(t)ρ+1 Eu(t)

ρ+2 ρ+2 D

  ρ+1  ≤ CEv(t)ρ+2 Eu(t)p ,

we have



1



1–α   ρ+1   1 1–α  1–α

E v(t) ρ v(t)u(t) dx ≤ Ev(t)ρ+2 E u(t)ρ+2

D

2  θ    ρ+2  (ρ+1) ζ   ≤ C Ev(t)ρ+2 (ρ+2)(1–α) + Eu(t)p 2(1–α) , where ζ1 + θ1 = 1. By choosing ζ = with (3.15), (3.36) becomes

(1–α)(ρ+2) (> 1), ρ+1

we have

θ 2(1–α)

=

ρ+2 2[(1–α)(ρ+2)–(ρ+1)]

1

 ρ+2



1–α 2 

  ρ+2  

E v(t) ρ v(t)u(t) dx ≤ C Ev(t)ρ+2 + Eu(t)p 2[(1–α)(ρ+2)–(ρ+1)] .

D

(3.36) < p2 . And

Kim et al. Boundary Value Problems (2018) 2018:14

Using Lemma 3.2 with s =

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ρ+2 , 2[(1–α)(ρ+2)–(ρ+1)]

we obtain



1



1–α  ρ+2  2 

E v(t) ρ v(t)u(t) dx ≤ C H(t) + Ev(t)ρ+2 + E∇v(t)

D

 p  2  + Eu(t)p + E(h ◦ ∇u)(t) + E∇u(t) .

Therefore, we deduce, for all t ≥ 0,  1 L 1–α (t) = H 1–α (t) +

1 1–α  

ρ δ E v(t) v(t)u(t) dx + δE ∇v(t) · ∇u(t) dx ρ +1 D D  ρ+2  2  ≤ C H(t) + Ev(t)ρ+2 + E∇v(t)  p  2  + Eu(t)p + E(h ◦ ∇u)(t) + E∇u(t) .

(3.37)

Combining (3.33) and (3.37) 1

L (t) ≥ KL 1–α (t),

∀t ≥ 0

with a positive constant K depending on C and δγ , it follows that 1–α

α

L 1–α (t) ≥

(1 – α)L

α – 1–α

(0) – αKt

.

Let T0 =

1–α α

αKL 1–α (0)

.

Then L(t) → ∞ as t → T0 . This means that there exists a positive time T ∗ ∈ (0, T0 ] such that   lim∗ E F(t) = +∞.

t→T

As for the case when P(τ∞ = +∞) < 1 (i.e., P(τ∞ < +∞) > 0), then ∇u(t) blows up in finite time T ∗ ∈ (0, τ∞ ) 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 ≥ p. We use the BorelCantelli lemma to prove the existence of a global solution. For this goal, we introduce an energy function ρ+2  2  2  p    e u(t) = ut (t)ρ+2 + ∇u(t) + ∇ut (t) + u(t)p + (h ◦ ∇u)(t).

(4.1)

T Theorem 4.1 Assume that (u0 , u1 ) ∈ H01 (D) × L2 (D), E 0 σ (t)2 dt < ∞, and condition (2.1) holds. If q ≥ p, u(t) is a solution of (1.1) with the initial data (u0 , u1 ) ∈ H01 (D) × L2 (D) according to Definition 2.1 on the interval [0, T], then for any T > 0, we have   E sup e u(t) < ∞. 0≤t≤T

(4.2)

Kim et al. Boundary Value Problems (2018) 2018:14

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Proof For any T > 0, we will show that uN (t) = u(t ∧ τN ) → u(t) (a.e.) 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 3. Similarly to Theorem 12 of [23], we can derive the proof of the theorem.  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). Abbreviations Stochastic quasilinear viscoelastic wave equation. Availability of data and materials No availability. 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. 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. Author details 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.

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