A general stability result for second order stochastic quasilinear evolution equations with memory

The goal of this paper is to discuss an initial boundary value problem for the stochastic quasilinear viscoelastic evolution equation with memory driven by additive noise. We prove the existence of global solution and asymptotic stability of the solution using some properties of the convex functions. Moreover, our result is established without imposing restrictive assumptions on the behavior of the relaxation function at infinity.


Introduction
The quasilinear viscoelastic wave equation of the following form: describes a viscoelastic material, with u(x, t) giving the position of material particle x at time t, where D is a bounded domain in R d with a smooth boundary ∂D, ρ > 0, g is the relaxation function, f denotes the body force, and h is the damping term. The properties of the solution to (1.1) have been studied by many authors (see [1][2][3][4][5][6][7]). For instance, in [1], Cavalcanti et al. considered (1.1) for h(u t ) = -γ u t and f (u) = 0, where 0 < ρ ≤ 2/(d -2) if d ≥ 3 or ρ > 0 if d = 1, 2. They proved a global existence result when the constant γ ≥ 0 and an exponential decay result for the case γ > 0. Messaoudi et al. [4] studied (1.1) for h(u t ) =u tt and f (u) = 0, they proved an explicit and general decay rate result with some properties of the convex functions. Liu [5] considered (1.1) for h(u t ) = 0 and f (u) = b|u| p-2 u, where b > 0, p > 2. The author proved that, for a certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. Conversely, he also obtained for certain initial data in the unstable set that there are solutions that blow up in finite time. In [6], Song investigated (1.1) for h(u t ) = |u t | q-2 u t and f (u) = |u| p-2 u, where q > 2, and ρ, p satisfy He proved the global nonexistence of the positive initial energy solutions of the quasilinear viscoelastic wave equation. Cavalcanti et al. [7] also studied (1.1) with h(u t ) = a(x)u t and f (u) = b|u| p-2 u, where a(x) can be null on a part of the boundary, they obtained an exponential rate of decay of solutions. In fact, the driving force may be affected by the random environment. In view of this, we consider the following stochastic quasilinear viscoelastic wave equations: where g is a positive function satisfying some conditions to be specified later, σ is local Lipschitz continuous, W (t, x) is an infinite dimensional Wiener random field, and the initial data u 0 (x) and u 1 (x) are F 0 -measurable given functions. To motivate our work, let us firstly recall some results regarding ρ = 0 and g ≡ 0, then (1.2) can be rewritten as the following stochastic wave equation: In [8,9], Chow considered the large-time asymptotic properties of solutions to a class of semi-linear stochastic wave equations with linear damping in a bounded domain. Under appropriate conditions, the author obtained the exponential stability of an equilibrium solution in mean-square and the almost sure sense by energy inequality. Using Lyapunov function techniques, Brzeźniak et al. [10] proved global existence and stability of solutions for the stochastic nonlinear beam equations. In [11], Brzeźniak and Zhu studied a type of stochastic nonlinear beam equation with locally Lipschitz coefficients. Using a suitable Lyapunov function and applying the Khasminskii test they showed the nonexplosion of the mild solutions. In addition, under some additional assumptions they proved the exponential stability of the solution. Kim [12] and Barbu et al. [13] investigated initial boundary value stochastic wave equations with nonlinear damping and dissipative damping, respectively. There are also many results on the stochastic wave equations, see the references in [10,[14][15][16][17][18][19][20]. When ρ = 0 and g = 0, (1.2) can be rewritten as the following stochastic viscoelastic wave equation: For the current equation (1.4), the memory part makes it difficult to estimate the energy by using these methods which are used in stochastic wave equation. Hence, Wei and Jiang [21] studied (1.4) with σ ≡ 1 and q = 2 in another way. They showed the existence and uniqueness of solution for (1.4) and obtained the decay estimate of the energy function of the solution under some appropriate assumption on g. In [22], Liang and Gao extended the existence and uniqueness results of [21] with σ = σ (u, ∇u, x, t). In the case of σ = σ (x, t), they proved that the solution either blows up in finite time with positive probability or is explosive in L 2 using the energy inequality. Furthermore, Liang and Gao [23] considered (1.4) driven by Lévy noise, they proved the global existence and uniqueness of the mild solution with the appropriate energy function and obtained the exponential stability of the solutions. Liang and Guo [24] studied (1.4) driven by multiplicative noise, the authors proved the global existence and asymptotic stability of the mild solution by the Lyapunov function.
We note that in the above literature, Messaoudi et al. [4], Liang and Gao [23], Chen et al. [24], and Kim et al. [25] did not discuss the optimality of the decay rate of (1.2) under the influence of random environment. We prove the stability of solutions to (1.2) by modifying the convex functions. The result of this paper provides an explicit energy decay formula that allows a larger class of functions g from which the energy decay rates are not necessarily of exponential or polynomial types. This paper is organized as follows. In Sect. 2, we present some assumptions and definitions needed for our work. Section 3 shows the statement and proof of our main result.

Preliminaries
Firstly, let us introduce some notations used throughout this paper. We set H = L 2 (D) with the inner product and norm denoted by (·, ·) and · 2 , respectively. Denote by ∇ · 2 the Dirichlet norm in V = H 1 0 (D). We consider the following hypotheses.
is a nonnegative and nonincreasing function satisfying There exists a positive function H ∈ C 1 (R+), with H(0) = 0, and H is a linear or strictly increasing and strictly convex C 2 function on (0, r] for some r < 1 such that , and such that (1.2) holds in the sense of distributions over (0, T) × D for almost all ω.
Similar to Theorem 4.1 of [25], we can explicitly drive the proof of the above theorem. Now, we introduce the "modified" energy associated with problem (1.2): where, for any w ∈ L 2 (D), Let (Ω, P, F) be a complete probability space for which {F t , t ≥ 0} of sub-σ -fields of F is given. A point of D will be denoted by D and E(·) stands for expectation with respect to probability measure P.
is a H-value Q-Wiener process on the probability space with the variance operator Q satisfying Tr Q < ∞. Moreover, we can assume that Q has the following form: are the corresponding eigenfunctions with c 0 := sup i≥1 e i ∞ < ∞ ( · ∞ denotes the super-norm). To simplify the computations, we assume that the covariance operator Q and -with homogeneous Dirichlet boundary condition have a common set of eigenfunctions, i.e., (2.5) and form an orthonormal base of H. In this case, For more details about the infinite-dimensional Wiener process and the stochastic integral, see in [26,27].

Stability properties of solutions
In this section, we state and prove our main stability result. Throughout this section, we As is well known, equation (1.2) is equivalent to the following Itô system: x).
In order to prove our stability result, we need the following lemmas.
Lemma 3.1 Let u 0 (x) and u 1 (x) be F 0 -measurable with u 0 (x) ∈ H 1 0 (D) and u 1 (x) ∈ L 2 (D). Assume (2.1) holds. Let (u, v) be a solution of system (3.2). Then we have d dt Proof Applying Itô's formula to 2 ρ+2 v ρ+2 ρ+2 , we get For the third term on the right-hand side of (3.4), we obtain Let From (3.1), we have satisfies, along the solution of (1.2), the estimate Proof Direct differentiation of Ψ , using (1.2), yields We take the expectation of the above formula to get the following result: for any general solution. With simple density parameters, this estimate is still applicable for weak solutions. Then we estimate that the second item on the right-hand side of (3.9) is as follows: By taking η = l 1-l , we obtain Inserting (3.11) in (3.9), we get (3.8).

Lemma 3.3 Let u be a solution of (1.2). The functional
satisfies the solution of equation (1.2) and, for any δ 1 , δ 2 > 0, the estimate Proof Differentiating (3.12) with respect to t and making use of (1.2), we arrive at We take the expectation of the above formula to get the following result: (3.14) Now we repeat the Cauchy-Schwarz inequality, Hölder's inequality and Young's inequality, to estimate each term on the right-hand side of equation (3.14). The first item on the right can be estimated as follows: As for the second item, we can get the following result from the previously obtained formula (3.10) and (a + b) 2 ≤ 2(a 2 + b 2 ): For the fourth term on the right-hand side of (3.14), it is easy to draw, ∀δ 2 > 0, For the fifth item, we can similarly get the following results: where C p is the Poincaré constant and δ 2 > 0. By using the Sobolev embedding and by (3.7), ∀t ≥ 0, we get Then (3.18) has the following form: Combining (3.14)-(3.17) and (3.21), we get (3.13). The proof is completed.

Theorem 3.4
Let u 0 (x) and u 1 (x) be F 0 -measurable with u 0 (x) ∈ L 2 (Ω; H 1 0 (D)) and u 1 (x) ∈ L 2 (Ω; L 2 (D)). Assume that (A1)-(A3) hold. Then there exist positive constants k 1 , k 2 , k 3 , and ε 0 such that the solution of (1.2) satisfies Moreover, if t 0 H 1 (t) dt < +∞ for some choice of J, then we have the improved estimate: 2. Our result is obtained under the very general assumption of the relaxation function g, which allows the processing of the larger class function g, which guarantees uniform stability of (1.2) and has a decay rate explicit formula energy. 3. The usual exponential and polynomial decay rate estimates have proven that g is satisfied (2.2) and g ≤ -kg p , 1 ≤ p < 3/2, it is a special case of our results. For these special cases, we will prove that this is a "simple" proof. 4. Our results allow the relaxation function to not necessarily exhibit exponential decay or polynomial decay. For example, if for 0 < q < 1 and a is chosen so that g satisfies (2.2), then g (t) = -H(g(t)) where, for t ∈ (0, r], r < a, which satisfies hypothesis (A3). Also, by taking J(t) = t α , (3.23) is satisfied with any α > 1. For this reason, we can use Theorem 3.4 and perform some calculations to infer that the energy is attenuated by the same g, i.e., 5. With (A2) and (A3), we can easily infer lim t→∞ g(t) = 0. This means that lim t→+∞ (-g (t)) cannot be equal to a positive number, so it is natural to assume lim t→+∞ (-g (t)) = 0. Therefore, there is t 1 > 0 big enough so that g(t 1 ) > 0 and max g(t), -g (t) < min r, H(r), H 0 (r) , ∀t ≥ t 1 . (3.25) As g is nonincreasing, g(0) > 0 and g(t 1 ) > 0, then g(t) > 0 for any t ∈ [0, t 1 ] and Hence, since H is a positive continuous function, then for some positive constants a and b. Consequently, for all t ∈ [0, t 1 ], which gives, for some positive constant d, we have Now let us prove Theorem 3.4.
It can be known from (A4) that it is obviously equivalent to EE(t). Therefore we have, for some a 0 > 0, F 0 (t) ≤ -a 0 F 1+α 0 (t), from which we easily infer that (3.33) By recalling that p < 3 2 and using (3.33), we find that +∞ 0 E(s) ds < +∞. Therefore, noting that Hence, repeating the above steps, with α = p -1, we obtain Thus the proof of Theorem 3.4 is completed.
This means that it is progressively stable and degenerates to (M + 2c)E 1 .