General decay of the double dispersive wave equation with memory and source terms

Shang [2] studied the well-posedness, asymptotic behavior, and the finite time blow-up of the solutions under some suitable conditions on f and for N = 1, 2, 3. Zhang and Hu [3] showed the existence and the stability of global weak solutions. Xie and Zhong [4] obtained the existence of global attractors in H1 0(Ω) × H1 0(Ω), where the nonlinear term f satisfies a critical exponential growth assumption. Xu et al., [5] used the multiplier method to investigate the asymptotic behavior of solutions for (3).

The motivation of our work is due to the initial boundary problem of the double dispersive-dissipative wave equation with nonlinear damping and source terms x ∈ Ω, a, b > 0, (2) which has been discussed by Di and Shang [1] by considering the existence of global solutions and the asymptotic behavior of global solutions with m ≥ p.
In the absence of the dispersive term and the nonlinear damping term, model (2) reduces to the following wave equation Shang [2] studied the well-posedness, asymptotic behavior, and the finite time blow-up of the solutions under some suitable conditions on f and for N = 1, 2, 3. Zhang and Hu [3] showed the existence and the stability of global weak solutions. Xie and Zhong [4] obtained the existence of global attractors in H 1 0 (Ω) × H 1 0 (Ω), where the nonlinear term f satisfies a critical exponential growth assumption. Xu et al., [5] used the multiplier method to investigate the asymptotic behavior of solutions for (3).
Mellah [6] considered the following initial-boundary value problem in a bounded domain and p > 1. He investigated the small data global weak solutions and general decay of solutions, respectively. Motivated by previous works, it is interesting to prove that problem (1) has a global weak solution assuming small initial data. In addition, we show the general decay of solutions. The global solutions are constructed by means of the Galerkin approximations and the general decay is obtained by employing the technique used in [7].

Preliminaries
In this section, we present some materials needed in the proof of our main result. We use the following abbreviations; · p = · L p (Ω) (1 ≤ p ≤ +∞) denotes usual L p norm, (·, ·) denotes the L 2 -inner product, and consider the Sobolev spaces H 1 0 (Ω) and H 2 0 (Ω) with their usual scalar products and norms. We also use the embedding In this case, the embedding constant is denoted by C * , that is u p ≤ C * ∇u 2 . We define By the direct computation, we deduce that Q is increasing Next, we give the assumptions for problem (1).
(G1) The relaxation function g : R + → R + is a bounded C 1 function such that (G2) There exist positive constants ξ 1 and ξ 2 such that (G3) We also assume that where λ 1 is the first eigenvalue of the following problem

Remark 1. [8]
Assuming λ 1 is the first eigenvalue of the problem (4), we have The energy associated with problem (1) is given by Now, we are in a position to state our main results.

Main results
In this section, we are going to obtain the existence of global weak solutions for problem (1) with the initial conditions ∇u 0 2 < z 0 and E(0) < Q(z 0 ).

Proof of Theorem 1 (Main result)
We divide the proof into two steps. In step 1, we prove the small data global existence of weak solutions by using the Faedo-Galerkin approximation and in step 2, we establish the general decay of energy employing the method used in [7].

Step 1: Global existence of weak solutions
Let ω j ∞ j=1 be an orthogonal basis of H 2 0 (Ω) with ω j being the eigenfunction of the following problem: Let V n = Span {ω 1 , ω 2 , · · ·, ω n }. By the standard method of ODE, we know that there exists only one local solution u n (t) = n ∑ j=1 b n j (t)ω j of the Cauchy problem as follows: u n (0) = u n 0 → u 0 , in H 2 0 (Ω), u n t (0) = u n 1 → u 1 in H 1 0 (Ω).
By the standard theory of ODE system, we prove the existence of solutions of problem (10)-(11) on some interval [0, t n ), 0 < t n < T for arbitrary T > 0, then, this solution can be extended to the whole interval [0, T] using the first estimate given below.

A Priori Estimates
Setting ω = u n t (t) in (10), we have A direct computation shows that Inserting (13) into (12) and integrating over [0, t] ⊂ [0, T], we obtain From assumption (G3) and the Sobolev embedding, we have u n p p ≤ C p * ∇u n p 2 , and then we have By using the fact that From E(0) < Q(z 0 ) and (11), it follows that for sufficiently large n. We claim that there exists an integer N such that Suppose the claim is proved, then Q( ∇u n 2 2 ) ≥ 0 and from (16) and (17), for sufficiently large n and 0 ≤ t < ∞.
Thus, we obtain that u is a global weak of problem (1). In order to prove (7), we use the mean value theorem, we see that there exists 0 < θ n < 1 such that and for each fixed t > 0, we obtain From (11), it follows that E n (0) → E(0) as n → +∞. Finally, taking n → +∞ in (14), we deduce that the energy identity (7) holds for 0 ≤ t < ∞.
Conflicts of Interest: "The author declares no conflict of interest."