*This work was support by the national research foundation of Korea (Grant NRF- 2016R1D1A1B03930361).
Let be a bounded domain with sufficiently smooth boundary , , , and have positive measures and
be the outward unit normal vector on . We denote , , where .
In this paper, we investigate the decay of energy of solutions for a von Karman system with memory and a delay term
where is assumed to satisfy if or if ,
; is a positive constant; is a real number; g is the kernel of the memory term; represents the time delay; are given functions belonging to suitable spaces; and the Airy stress function v satisfies the following elliptic problem
The von Karman bracket is given by
here is Poisson’s ratio,
From the physical point of view, problem (1) describes small vibrations of a thin homogeneous isotropic plate of uniform thickness of ; denotes the transversal displacement of the plate; the Airy stress function is a vibrating plate.
When and , problem (1) was studied by many authors  -  . The authors in    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     . 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  -  and references therein. Considering delay term , the problem is different from existing literature. Time delays arise in many applications depending not only on the present state but also on some past occurrences. And the presence of delay may be a source of instability (see e.g.   ). Thus, recently, the control of partial differential equations with time delay effects has become an active area of research (see     and references therein). Nicaise and Pignotti  examined a wave equation with a time-delay of the form
They proved that the energy of the problem decays exponentially under the condition
and there exists a sequence of delays such that instability occurs in the case . Kirane and Said-Houari  considered a viscoelastic wave equation with a delay
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 owing to the presence of the viscoelastic damping. Recently, Wu  obtained similar decay results as in  for problem (1) without von Karman bracket 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  to a nonlinear viscoelastic von Karman equation without the assumption . 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.
2. Statement of Main Results
Throughout this paper, we denote
For a Banach space X, denotes the norm of X. For simplicity, we denote by and by , respectively. We define for all
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.
For , the bilinear form is defined by
A simple calculation, based on the integration by parts formula, yields
Thus, for it holds
Since we know (see e.g.  ) that is equivalent to the norm on W, i.e.
This and Sobolev imbedding theorem imply that for some positive constants , and
By (7) and Young’s inequality, we see that
From this and (8), it holds that
We introduce the relative results of the Airy stress function and von Karman bracket .
Lemma 2.1. (  ) If and belong in and at least one of them belongs in , then .
Lemma 2.2. (  ) Let and be the Airy stress function satisfying (2). Then, the following relations hold:
Now, we state the assumptions for problem (1).
(H1) For the relaxation function g, as in   , we assume that is a nonincreasing differentiable function satisfying ,
where is a nonincreasing differentiable function.
Theorem 2.1. Assume that (H1) is hold. Then, for the initial data problem (1) has a unique weak solution in the class
Proof. This can be proved by Faedo-Galerkin method (see e.g.   ).
3. 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
From (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 , we have
From the symmetry of , we see that for any
Moreover, (10) gives
Now, we define a modified energy by
where p is a positive constant satisfying
It is noted that . Therefore, it is enough to obtain the desired decay for the modified energy which will be done below.
Lemma 3.1. There exist non-negative constants and satisfying
Proof. Applying (14) to the last term in the right hand side of (13), we have
By Young’s inequality,
Thus, we have
Putting , and considering (16), we complete the proof. ,
Now, let us define the perturbed modified energy by
Then, it is easily shown that is equivalent with for all . ,
Lemma 3.2. There exist positive constants and satisfying
Proof. Poincare’s inequality gives
where is the embedding constant from to . Using the problem (1) and (14), we have
Young and Poincaré’s inequalities produce
Substituting these into (21), we derive
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  , we have
Using Young inequality and the fact that imbedding is continuous, we infer
where is the embedding constant from to .
Young’s inequality and (10) give
Combining these estimates with (23), we get
Since g is positive, for any we have for all . Thus, combining (17), (22) and (24), we arrive
First, we fix and such that and , respectively. Next, we choose so large that , and sufficiently small such that , and . Finally, taking so small that , and , we complete the proof.,
Theorem 3.1. There exist positive constants and such that
Proof. Multiplying (18) by , using (11) and (17), we get
Since is nonincreasing, we have
Thus, by letting , we get
Since is a nonincreasing positive function, we can easily observe that is equivalent to . Subsequently, it follows that
Integrating this over , we conclude that
Consequently, the equivalent relations of and yield the desired result. ,
In this paper we proved decay rates of energy for a viscoelastic von Karman equation with constant time delay in the velocity by establishing proper Lyapunov functionals corresponding to the delay effect. In the future work, we will consider the equation with time-varying delay effect.
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