Energy Decay for a Von Karman Equation of Memory Type with a Delay Term

We consider a von Karman equation of memory type with a delay term ( ) ( ) ( ) [ ] 2 2 0 1 0 d , , . t t tt tt t t u u u u g t s u s s a u a u x t u v ρ α τ − ∆ + ∆ − − ∆ + + − = ∫ By introducing suitable energy and Lyapunov functional, we establish a general decay estimate for the energy, which depends on the behavior of g.

Γ Γ ≠ ∅ ∩ , 0 Γ and 1 Γ have p ositi v e mea sur es and , ν ν ν = be the outward unit normal vector on ∂Ω .We denote t u u t ( ) In this paper, we investigate the decay of energy of solutions for a von Karman system with memory and a delay term , , in 0, , 0 on 0, , d 0 on 0, , where ρ is assumed to satisfy a is a positive constant; 1 a is a real number; g is the kernel of the memory term; 0 τ > represents the time delay; 0 1 0 , , u u f are given functions belonging to suitable spaces; and the Airy stress function v satisfies the following , in 0, , 0 on 0, ).
From the physical point of view, problem (1) describes small vibrations of a thin homogeneous isotropic plate of uniform thickness of α ; ( ) denotes the transversal displacement of the plate; the Airy stress function ( ) When 1 0 a = and 0 ρ = , problem (1) was studied by many authors [1]- [8].
The authors in [1] [3] [4] proved uniform decay rates for the von Karman system with frictional dissipative effects in the boundary.The stability for a von Karman system with memory and boundary memory conditions was treated in [5] [6] [7] [9].They proved the exponential or polynomial decay rate when the relaxation function decay is at the same rate.The aim of this work is to prove a general decay result for a nonlinear von Karman equation of memory type with a delay term in the first equation of (1), when the relaxation function does not necessarily decay exponentially or polynomially.As for the works about general decay for viscoelastic system, we refer [10]- [15] and references therein.
Considering delay term ( ) The authors proved the existence of a solution and a general decay result under the condition They showed that the energy of solutions is still asymptotically stable even if a a = owing to the presence of the viscoelastic damping.Recently, Wu [20] obtained similar decay results as in [21] for problem (1) without von Karman bracket [ ] , u v under the condition (5).Motivated by these results, we prove a general decay result for a nonlinear viscoelastic von Karman Equation (1) with a time-delay under the condition which is an extension and improvement of the previous result from [20] to a nonlinear viscoelastic von Karman equation without the assumption 1 0 a > .
The plan of this paper is as follows.In Section 2, we give some notations and materials needed for our work.In Section 3, we derive general decay estimate of the energy.

Statement of Main Results
Throughout this paper, we denote , d and , d .
For a Banach space X, X ⋅ denotes the norm of X.For simplicity, we denote From now on, we shall omit x and t in all functions of x and t if there is no ambiguity, and c denotes a generic positive constant different from line to line or even in the same line.
This and Sobolev imbedding theorem imply that for some positive constants , , , and , , .
By ( 7) and Young's inequality, we see that H Ω and at least one of them belongs in ( ) ∈ Ω and v be the Airy stress function satisfying (2).Then, the following relations hold: , and , .
Now, we state the assumptions for problem (1).
(H1) For the relaxation function g, as in [11] [15], we assume that : where : Theorem 2.1.Assume that (H1) is hold.Then, for the initial data problem (1) has a unique weak solution u in the class Proof.This can be proved by Faedo-Galerkin method (see e.g.[7] [21]).

General Decay of the Energy
In this section we shall prove a general decay rate of the solution for problem (1).For simplicity of notations, we denote 9), we see that From now on, we shall omit t in all functions of t if there is no ambiguity, and c denotes a generic positive constant different in various occurrences.Multiplying the first equation of (1) by t u , we have where From the symmetry of ( ) , a ⋅ ⋅ , we see that for any ( ) ( ) Moreover, (10) gives Now, we define a modified energy by Proof.Applying (14) to the last term in the right hand side of (13), we have and considering ( 16), we complete the proof. Now, let us define the perturbed modified energy by where 1 λ is the embedding constant from ( ) L Ω .Using the problem ( 1) and ( 14), we have Young and Poincaré's inequalities produce Similarly, we get from (1) that In what follows we will estimate the terms in right hand side of (23).By similar arguments given in [8], we have ( ) Ω is continuous, we infer where 2 λ is the embedding constant from V to ( ) ( ) Young's inequality and (10) give Since g is positive, for any 0 0 t > we have ( ) ( ) for all 0 t t ≥ .Thus, combining ( 17), ( 22) and (24), we arrive

Conclusion
In this paper we proved decay rates of energy for a viscoelastic von Karman