Blow-Up for a Stochastic Viscoelastic Lamé Equation with Logarithmic Nonlinearity

Mascara University, Faculty of Exact Sciences, Mascara 29000, Algeria Mascara University, Faculty of Economies Sciences, Mascara 29000, Algeria Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria Preparatory Institute for Engineering Studies in Sfax, Tunisia


Introduction
In recent years, stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, blow-up, and other quantitative and qualitative properties of solutions have been established.
In this work, we consider the following problem of stochastic wave equation: where D is a bounded domain in IR n , n ∈ IN * , with a smooth boundary ∂D; μ, λ are the Lamé constants which satisfy μ > 0 , λ + μ ≥ 0; h is a positive function, p > q ≥ 2; the constant k is a small nonnegative real number; and L 2 ðDÞ is the set of square integrable function on D equipped with the inner product h:, :i and its norm k:k 2 .
Wðx, tÞ is an infinite dimensional Wiener process, σðx, tÞ is L 2 ðDÞ valued progressively measurable, and ε is a positive constant which measures the strength of noise.
It is common to observe a wave motion as a physical phenomenon which is mathematically modeled by a partial differential equation of hyperbolic type. Much has been written about such equations regarding their widespread applications to engineering and sciences. However, for more realistic models, the random fluctuation had been taken into consideration which led to introduced stochastic wave equation in 1960's. Several examples of linear stochastic wave propagation and applications can be found in [1]. Mueller [2] was the first who investigate the existence of explosive solutions for some stochastic wave equation. Motivated by Mueller [2], Chow [3] was interested by knowing how does a random perturbation affect the solution behavior for a wave equation with a polynomial nonlinearity. He was concerned with the existence of local and global solutions of the stochastic equation: where the initial data g and h are given functions and the nonlinear terms f ðuÞ and σðuÞ are assumed to be polynomials in u. Four years later, he [4] established an energy inequality and the exponential bound for a linear stochastic equation and gave the existence theorem for a unique global solution for the randomly perturbed wave equation: In 2009, Chow [5] studied the problem of explosive solutions for a class of nonlinear stochastic wave equation in a domain D ⊂ ℝ d for d ≥ 3, We can mention some other works such as Cheng et al. [6] who studied the existence of a global solution and blowup solutions for the nonlinear stochastic viscoelastic wave equation with nonlinear damping and source terms: The authors proved that finite time blow-up with nonnegative probability is explosive or it is explosive in energy sense for p > q.
Moreover, Kim et al. [7] considered the stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms: They showed the existence of a global solution and blowup in finite time.
Recently, Yang et al. [8] treated the following stochastic nonlinear viscoelastic wave equation: Journal of Function Spaces They established the existence of global solution and asymptotic stability of the solution by using some properties of the convex function.
However, it was noticed that the logarithmic nonlinearity appears naturally in many branches of physics such as nuclear physics, optics, and geophysics (see [9,10]). These specific applications in physics and other fields attract a lot of mathematical scientists to work with such problems. In the deterministic case, Al-Gharabli [11] investigated the stability of the solution of a viscoelastic plate equation with a logarithmic nonlinearity source term for the following problem: where D ⊆ ℝ 2 is a bounded domain with a smooth boundary ∂D. The vector ν is the unit outer normal to ∂D, and h is the nondecreasing nonnegative function. Mezouar et al. [12] treated a more general problem where they considered the following nonlinear viscoelastic Kirchhoff equation with a time-varying delay term: The paper is organized as follows: in Section 2, we introduce some basic definitions, necessary assumptions, and lemmas that are helpful in proving our main result. Section 3 is devoted to show the blow-up of the solution of our problem.

Preliminaries
Let ðΩ, F, PÞ be a complete probability space for which a filtration fF t , t ≥ 0g of increasing sub σ − fields F t is given and Wðx, tÞ be a continuous Wiener random field in this space with a mean zero and the covariance operator Q satisfying Wðx, tÞ is defined by where β j ðtÞ is a sequence of real-valued standard Brownian motions mutually independent on the probability space ðΩ , F, PÞ, λ j are the eigenvalues of Q, and e j are the corresponding eigenvectors. That is, Note Eð:Þ stands for expectation with respect to probability measure P. Let H be the set of L 0 2 = L 2 ðQ 1/2 V, VÞ-valued processes with the norm where ϕ * ðsÞ denotes the adjoint operator of ϕðsÞ and V = H 1 0 ðDÞ which is equivalent to H 1 ðDÞ. For any process ϕðsÞ ∈ H , we can define the stochastic integral with respect to the Q-Wiener process as Ð t 0 ϕðsÞdWðsÞ which is a martingale. For more details about the infinite dimension Wiener process and stochastic integral, we refer to Da Prato and Zabczyk (pp. 90-96, [13]).
To state and prove our result, we need some assumptions. A1. Assume that h : and there exist tow nonnegative constants ς 1 and ς 2 such that 3 Journal of Function Spaces A2.
The following theorem states the existence and uniqueness of a local solution of our problem; the proof can be established by combining the proof given in [6,12].
We define the energy associated to the solution of system (1) by where hov We rewrite (1)

Journal of Function Spaces
Then there exists a constant C s = C s ðD, mÞ such that Lemma 3 [15]. For h, φ ∈ C 1 ð½0,+∞½, IRÞ, we have Proof. We can apply the Itô's formula to (21) for each x ∈ D after integrating the above equation over D to get By using integration by parts, we get By applying Lemma 3, we have We have By replacing (29)-(32) in (28) and multiplying equation (28) by 1/2, we arrive at (27).

Blow-Up
We prove our main result for p > q; we purpose where 5 Journal of Function Spaces Lemma 5. Let ðu, vÞ be a solution of system (21) with initial data ðu 0 , v 0 Þ ∈ H 1 0 ðDÞ × L 2 ðDÞ. Then, we have Proof. Using the Itô's formula and by following the same way as our discussions in Lemma 4 with taking the expectations, we obtain (37). We multiply the second equation in (22) by u and integrate the result over D, and we take expectation; we obtain (38).
We set HðtÞ = GðtÞ − EeðtÞ: As h is a positive decreasing function so Consequently, Lemma 6. Let ðu, vÞ be a solution of system (21). Assume that (A1) holds. Then, there exists a positive constant C such that where 2 ≤ s ≤ p + 1. Proof.
The last inequality is getting from (A1).
We are ready to state and prove our main result for p > q. For this purpose, we define where 0 < α < min and δ is a very small constant determined later. 6 Journal of Function Spaces Theorem 9. Assume (A1) and (A2) hold. Let ðu, vÞ be a solution of system (21) with initial data ðu 0 , v 0 Þ ∈ H 1 0 ðDÞ × L 2 ðDÞ satisfying where β is a nonnegative constant and E 1 is given in (35). If p > q, then there exists a positive time T 0 ∈ ½0, T such that where and K is given later.
Proof. Let A direct differentiation of LðtÞ gives Recalling (39) and (19), (51) leads to Hence, where ξ and c are constants. We consider the following partition of D: