On Decay of Solutions to Systems of Integro-differential Equations with Strongly Damped

We study the system of nonlinear integro-differential equations with strong damping and weak damping terms, in a bounded domain with the initial and Dirichlet boundary conditions. The existence of global solutions by using the potential well method, and the energy decay estimate by applying a lemma of Komornik [3]


Introduction
In this paper we consider the following initial-boundary value problem where Ω is a bounded domain with smooth boundary ∂Ω in R n ; f i (., .) : R 2 −→ R are given functions to be specified later.
A single wave equation of the problem (1) becomes as following 26 On Decay of Solutions to Systems of Integro-differential Equations with Strongly Damped The global existence and blow up of solution for (2) were established [7]. In the absence of the dispersive term △u tt and the weak damping term u t , were established [8]. In the absence of the dispersive term △u tt and the strong damping term △u t , were established [11,9]. Also, Liang and Gao [12] studied the global existence, decay and blow up of solution problem (2) with the absence of the dispersive term △u tt .
Liang and Gao [10] studied the global existence, decay and blow up of solution problem (1) with the absence of the dispersive term △u tt and the weak damping term u t .
In this paper, under some restrictions on the initial data, we establish the global existence and the decay of solutions.
This paper is organized as follows. In section 2, we present some lemmas, and the local existence theorem. In section 3, the global existence and energy decay of the solution are given.

Preliminaries
In this section, we shall give some assumptions and lemmas which will be used throughout this work. Let ∥.∥ and ∥.∥ p denote the usual L 2 (Ω) norm and L p (Ω) norm, respectively. First, we make the following assumptions: Concerning the functions f 1 (u, v) and f 2 (u, v) , we take where k, l > 0 are constants and r satisfies    −1 < r if n = 1, 2, According to the above equalities they can easily verify that where We have the following result.
Lemma 1 [5]. There exist two positive constants c 0 and c 1 such that is satisfied.
Let's define and also the energy function as follows where Lemma 2 E (t) is a nonincreasing function for t ≥ 0 and Proof. The proof is almost the same that of [5], so omit it here.
Moreover, the following energy inequality holds: Lemma 3 (Sobolev-Poincare inequality) [1]. Let p be a number with 2 ≤ p < ∞ (n = 1, 2) or 2 ≤ p ≤ 2n/ (n − 2) (n ≥ 3) , then there is a constant C * = C * (Ω, p) such that The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1).

Then we have
Next, we state the local existence theorem that can be established by combining arguments of [2,4].
Moreover, at least one of the following statements holds true:

Remark 6
We denote by C various positive constants which may be different at different occurrences.

28
On Decay of Solutions to Systems of Integro-differential Equations with Strongly Damped

Global existence and energy decay
In this section, we consider the global existence and energy decay of solutions for problem (1).
For the sake of simplicity and to prove our result, we take k = l = 1 and introduce where η is the optimal constant in (11). Next, we will state and prove a lemma which is similar to the one introduced firstly by Vitillaro in [6] to study a class of a single wave equation. Then for all t ∈ [0, T ) . (8), (9), (11) and the definition of B, we have

Proof. First from
So, we get where . Note that G (α) has the maximum at α * = B − r+2 r+1 and maximum value is Now we establish (14) by contradiction. Suppose (14) does not hold, then it follows from the continuity of (u (t) , v (t)) that there exists t 0 ∈ (0, T ) such that ( By (15), we see that (14) is established.
then we have the following decay estimates it follows from Lemma 8 that which implies that (7) and (8), we get ) .
Next, we want to derive the decay rate of energy function for problem (1). Multiplying the first equation of system (1) by u and the second equation of system (1) by v, integrating them over Ω × [t 1 , t 2 ] (0 ≤ t 1 ≤ t 2 ) , using integration by parts and summing up, we have ∫ It follows from (9) 2 For the eleventh term on the right hand side of (25), we have and similarly, we have Substituting (26), (27) into (25), we have In what follows we will estimate A 1 , A 2 , ..., A 8 in (28). Firstly, by Hölder, Young and Sobolev Poincare inequalities, we have Then, by (24), we have For A 2 in (28), applying ∥u t ∥ 2 + ∥v t ∥ 2 ≤ −E ′ (t) from (10), we have Similarly, we have and 32 On Decay of Solutions to Systems of Integro-differential Equations with Strongly Damped We also have the following estimate Similarly Using Young inequality for convolution ∥f * g∥ 1 ≤ ∥f ∥ 1 ∥g∥ 1 , we have From (23), we have Combining (35) and (36), we have Inserting estimates (29)-(37) into (28), we arrive at where C 7 = 2C 1 + 2C 2 + 2C 3 + 2C 4 + C 5 + C 6 .
Since µ > 0 from the assumption of conditions, by Lemma 4, we have for every t ≥ C7 µ . The proof is completed.