Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

Abdelmajid Ali Dafallah1,∗, Fadlallah Mustafa Mosa2, Mohamed Y. A. Bakhet3 and Eshag Mohamed Ahmed4 1 Faculty of Petroleum and Hydrology Engineering, Alsalam University, Almugled, Sudan. 2 Department of Mathematics and physics, Faculty of Education University of Kassala Kassala, Sudan. 3 Department of Mathematics, College of Education Rumbek University of Science and Technology Rumbek, South Sudan. 4 Faculty of Pure and Applied Sciences, International University of Africa, Khartoum, Sudan. * Correspondence: majid_dafallah@yahoo.com

The basic concepts and notions of random attractors for the infinite dimensional was recently presented by in [6][7][8][9]. A random attractor of RDS is a measurable and compact invariant random set attracting all orbits. whilst such an attracting set exists, it is the smallest attracting compact set and the largest invariant set. In recent years, a random attractor for autonomous and non-autonomous stochastic dynamical systems have been studied by many authors, see for example [10][11][12][13][14][15][16] and the references therein.
In the deterministic case; that is, κ = 0 in (1), the asymptotic behavior of the solution for global attractors an extensible beam equation with localized nonlinear damping with memory has been studied in [5,[17][18][19].
In [20], for the case of µ = 0 in (1), the authors investigated the existence of random attractor for the stochastic an extensible beam equation with localized nonlinear damping without memory. But, there were no results even for the bounded case. While it is far just our interest in this paper. To the best of our knowledge, the dynamics of system (1) involving but essential difficulties in showing compactness by using the uniform estimates on the tails of solution. Motivated by a similar technique of [16].
The rest of the paper is organized as follows. In Section 2, we recall some basic concepts and properties for general random dynamical systems. In Section 3, we first provide some basic settings about (1) and show that it generates a random dynamical system in proper function space. In Section 4, we prove the existence of a unique random attractor of the random dynamical system by bounded absorbing set and using a compact measurable pullback attracting set.

Preliminaries
In this section, we recall some basic concepts related to random attractors for stochastic dynamical systems. The readers are referred to [6][7][8] for more details. Which are crucial for getting our main results. Let (Ω, F , P) be a probability space and (X, d) be a Polish space with the Borel σ-algebra B(X). The distance between x ∈ X and B⊆ X is denoted by d(x, B). If B⊆ X and C⊆ X, the Hausdorff semi-distance from B to C is denoted by way of d(B, C) = sup x∈B d(x, C).
for all s,t ∈ R and θ 0 P = P for all t∈ R.
Definition 5. Let D be a collection of random subset of X and K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D, then K is called an absorbing set of Φ ∈ D if for all τ ∈ R, ω ∈ Ω and B ∈ D, there exists, T = T(τ, ω, B) > 0 such that Definition 6. Let D be a collection of random subset of X, the Φ is said to be D-pullback asymptotically compact in X if for p-a.e ω ∈ Ω , {Φ(t n , θ −t n ω , x n )} ∞ n=1 has a convergent subsequence in X when t n → ∞ and x n ∈ B(θ −t n ω) with {B(ω)} ω∈Ω ∈ D. Definition 7. Let D be a collection of random subset of X and A = {A(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D, then A is called a D-random attractor (or D-pullback attractor) for Φ, if the following conditions are satisfied for all t ∈ R + , τ ∈ R and ω ∈ Ω where d X is the Hausdorff semi-distance given by d X (Y, Z) = sup y∈Y inf z∈Z y − z X for any Y ∈ X and Z ∈ X.
Lemma 1. Let D be a neighborhood-closed collection of (τ, ω)-parameterized families of nonempty subsets of X and Φ be a continuous cocycle on X over R and (Ω, , P, (θ t ) t∈R ). Then Φ has a pullback D-attractor A in D if and only if Φ is pullback D-asymptotically compact in X and Φ has a closed, -measurable pullback D-absorbing set K ∈ D, the unique pullback D-attractor A = A(τ , ω) is given

Existence and uniqueness of solution
In this Section, first, we collect some important results that will help to achieve our goal. Let A = ∆ 2 , A 1 2 = −∆ and D(A) = {u ∈ H 4 : ∆ ∈ H 1 0 . We can define the powers A ν is Hilbert space and a norm hold We denote that the injection V ν 1 → V ν 2 is compact embeddings, if ν 1 > ν 2 in conjunction with the generalized Poincaré inequality; where λ 1 is the first eigenvalue of A. Additionally we outline the subsequent Much like [18], for the memory kernel hypotheses µ(·), we suppose L 2 µ (R + ; V ν ) the Hilbert space of function η : R + −→ V ν endowed with the inner product and norm respectively, specially, u 2 µ,ν = u 2 µ,1 . Let, we define the product Hilbert space To convert the version of Problem (6) with a random perturbation term right into a deterministic one with a random parameter ω, we introduce an Ornstein-Uhlenbeck process driven by means of the Brownian motion, which satisfies the subsequent differential equation Its unique stationary solution is given by From [6,16], it is recognized that the random variable |z j (ω j )| is tempered and there is an invariant set Ω ⊆ Ω of full P measure such that z j (θ t ω j ) = z j (t, ω j ) is continuous in t, for each ω ∈Ω. For comfort, we shall writeΩ as Ω. It follows from Proposition 3.4 in [16], that for any > 0, there exists a tempered characteristic (ω) > 0 such that where (ω) satisfies for, p-a.e. ω ∈ Ω, Then, it follows from the above inequality, for p-a.e. ω ∈ Ω, Put , we handy to reduce (6) to an evolution equation of the first-order in time random partial differential equation (RPDE): (16) Consequently the stochastic system for the system (16) becomes [21], we have the fact that H is the infinitesimal generators of C 0 -semigroup e Ht on E(Γ). It is not difficult to check that the function Q(ψ, ω, t) : E → E is locally Lipschitz continuous with respect to ψ and bounded for each ω ∈ Ω.

Random absorbing set
In this section, we will show boundedness of the solutions for Equation (17). The existence of a pullback absorbing set Φ ∈ D and the asymptotic compactness of the random dynamical system associated with the Equation (17). We always assume that D is the collection of all tempered subsets of E(Γ) from now on. (2)-(4) and (7)-(8) hold. Then, for any τ ∈ R, ω ∈ Ω and χ τ−t ∈ E(Γ), there exists a random ball {K(ω)} ω∈Ω ∈ D centered at 0 with random radius M(ω) ≥ 0 such that {K(ω)} is a random absorbing set for Φ in D, that is, for any B={B(ω)} ω∈Ω ∈ D, P-almost surely, there exists a T = T(τ, ω, B) > 0 and χ τ−t (ω) ∈ B(ω) such that

Lemma 2. Let
where M 0 (ω) is a positive random function, that is Proof. Taking the inner product of the first term of (23) with Using Hölder, Young and Poincarè inequalities and after simple computation, we gain Using Cauchy-Schwartz inequality and Young inequality, we obtain and from (2) and (4), it is easy to show that where ϑ is between 0 and v − εu By second term for right hand side of (26) and (29), we can get About the nonlinearity, by (4), Hölder inequality and the Sobolev embedding theorem, we estimate that Inserting the above two inequalities together, it yields that (37) Collecting all inequalities (25)-(37), it leads to d dt Thus and Since ε ∈ (0, 1) be small enough such that ε 2 m − Applying Gronwall's Lemma over [τ − t , r], we find that for r ≥ τ − t, By replacing ω by θ −t ω, we get from (42) such that for all t ≥ 0 Since z(θ t ω) is a tempered random variable and lim t→±∞ z(θ t ω) t = 0, 0 ±∞ 1 t z(θ r ω)dr = 0. Thus, there exists M 0 (ω) and T = T(τ, ω, B) > 0 such that The proof is completed. Now we decompose the Equation (6) into two parts and also decompose the nonlinear growth term f ∈ C 1 in Equation (3) into two parts f = f 1 + f 2 , where f 1 , f 2 satisfy the following respectively where µ i , C µ , k 0 , k 1 , i = 1, 2 are positive constants. Let for any τ ∈ R, ω ∈ Ω, there is a time for any ω ∈ Ω, whereT =T(B 0 , ω) ≥ τ is the pullback absorbing time in Lemma 2, then it holdsB(ω) ⊆ B 0 (ω) that In order to obtain the regularity estimates, we decompose the solution Then, we can rewrite the Equation (6) into the following systems Letχ(t, ω) = (ŷ,y,η t (t, s)) ,ŷ = y andy =ŷ t + εŷ, which are equivalent with where The above equations leads to in which Now we need to establish some priori estimates for the solutions of Equation (50) and Equation (53), which are the basis of our later analysis.  B(τ, ω). Then there existsT =T(B, ω) > 0 and M 0 (ω), such that the solutionχ(T, ω,χ τ (ω)) of (50) satisfies for P-a.e ω ∈ Ω, ∀ t ≥T Proof. Taking inner product of (50) withχ in E, we have Using Hölder, Young and Poincarè inequalities, we get Now, we estimate the terms on the right hand side of (55) one by one: and from (2), it is easy to show that (a(x)g(ŷ t ),y) = (α 0 g(ϑ) (y − εŷ,y)) ≤ α 0 α 1 y 2 − α 0 α 2 ε (ŷ,y) , where ϑ is between 0 andy − εŷ.
Now we obtain our main result about the existence of a random attractor for random dynamical system Φ as following Lemma. It follows from Lemma 2, that Φ has a closed random absorbing set in D, then apply Lemmas in Section 4, we prove the existence of a random attractor by using tail estimates and the decompose technique of solutions. which along with the D-pullback asymptotic compactness. Lemma 5. (see [2,3,15]) Let X 0 , X, X 1 be three Banach spaces such that X 0 → X → X 1 is projection operator X 0 → X is compact. setting Y = χ(t,B(τ, ω)) ⊂ L 2 µ (R + , X) is a random bounded absorbing set from Lemma 4, ψ(t) is the solution operators of (53) and by Lemma 4, there is a positive random radius M ν (ω) dependent on t, such that 1). Y is bounded in L 2 µ (R + , X 0 ) H 1 µ (R + , X 1 ), 2). sup η∈Y,s∈R + ∇η(s) 2 X ≤ M ν (ω).