General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping

1 Introduction

Let ℋ = Ω × ( τ 1 , τ 2 ) × ( 0 , ∞ ) , in the present work, we consider the following wave equation:

where Ω ∈ R N is a bounded domain with sufficiently smooth boundary ∂ Ω . ζ 0 , ζ 1 , σ , β 1 are positive constants, m ≥ 1 for N = 1 , 2 , and 1 < m ≤ N + 2 N − 2 for N ≥ 3 .

τ 1 < τ 2 are non-negative constants such that β 2 : [ τ 1 , τ 2 ] → R represents distributive time delay, h , α are positive functions.

The importance of the viscoelastic properties of materials has been realized because of the accelerated development of the rubber and plastics industry. Physically, the viscoelastic damping term is the relation-ship between the stress and strain history in the beam inspired by the Boltzmann theory, where the kernel of the term of memory is the function h , see [1,2, 3,4,5, 6,7].

In [8], Balakrishnan and Taylor proposed a new model of damping and called it the Balakrishnan-Taylor damping, as it relates to the span problem and the plate equation. For more details, the readers can refer to some papers that focused on the study of this damping [9,10,11, 12,13].

On the other hand, the stability issue of systems with delay is of theoretical and practical great importance, whereas, the dynamic systems with delay terms have become a major research subject in differential equation since the last five decades. Recently, the stability and the asymptotic behavior of evolution systems with time delay especially the distributed delay effect have been studied by many authors, see [14,15,16, 17,18].

Very recently, in [19] the authors considered our problem (1.1) but in the presence of the delay, they proved the general decay result of solutions by the energy method under suitable assumptions.

Based on all of the above, the combination of these terms of damping (memory term, Balakrishnan-Taylor damping, and the distributed delay ) in one particular problem with the addition of α ( t ) to the term of memory and the distributed delay term ∫ τ 1 τ 2 ∣ β 2 ( s ) ∣ ∣ u t ( t − s ) ∣ m − 2 u t ( t − s ) d s we believe that it constitutes a new problem worthy of study and research, different from the above that we will try to shed light on.

Our paper is divided into several sections: in Section 2 we lay down the hypotheses, concepts, and lemmas we need, and in Section 3 we prove our main result. Finally, we give a conclusion in Section 4.

2 Preliminaries

For studying our problem, in this section we need some materials.

First, introducing the following hypothesis for β 2 , h , and α :

1. h , α : R + → R + are non-increasing C 1 functions satisfying

   (2.1) h ( t ) > 0 , α ( t ) > 0 , l 0 = ∫ 0 ∞ h ( ϱ ) d ϱ < ∞ , ζ 0 − 2 α ( t ) ∫ 0 t h ( ϱ ) d ϱ ≥ l > 0 ;

2. ∃ ϑ : R + → R + is a non-increasing C 1 function satisfying

   (2.2) ϑ ( t ) h ( t ) + h ′ ( t ) ≤ 0 , t ≥ 0 and lim t → ∞ − α ′ ( t ) ϑ ( t ) α ( t ) = 0 ;

3. β 2 : [ τ 1 , τ 2 ] → R is a bounded function satisfying

   (2.3) ∫ τ 1 τ 2 ∣ β 2 ( s ) ∣ d s < β 1 .

and

As in [18], taking the following new variables:

which satisfy

So, problem (1.1) can be written as

where

Now, we give the energy functional.

Now we state the local existence of problem (2.5).

3 General decay

In this section, we state and prove the asymptotic behavior of system (2.5). For this goal, we set

and

and