Optimal polynomial decay for a coupled system of wave with past history

Abstract

This work deals with a coupled system of wave with past history effective just in one of the equations. We show that the dissipation given by the memory effect is not strong enough to produce exponential decay. On the other hand, we show that the solution of this system decays polynomially with rate t − 1 2 . Moreover by recent result due to A. Borichev and Y. Tomilov, we show that the rate is optimal. To the best of our knowledge, there is no result for optimal rate of polynomial decay for coupled wave systems with memory in the previous literature.

Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface

“…They proved that the solution of the system (1.13) decays polynomially with rate t − 1 2 . Also, in 2020, Cordeiro et al 23 established the optimality of the decay rate. But to the best of our knowledge, it seems that no result in the literature exists concerning the case of coupled wave equations with localized past history damping, especially in the absence of smoothness of the damping and coupling coefficients.…”

“…Remark 5.1. In Almeida et al 22 and Cordeiro, 23 the authors proved the lack of exponential stability of a coupled wave equations system with past history damping by taking a particular relaxation function g(s) = e − s such that s ∈ R + and > 1.…”

Optimal polynomial decay for a Timoshenko system with a strong damping and a strong delay

“…To demonstrate the lack of exponential stability, we will employ the above theorem integrated with some techniques used in [22][23][24][25][26] taking into account the nature of our problem. So, we will prove that there exists a subsequence (…”

Exponential stability for a piezoelectric beam with a magnetic effect and past history

“…However, in the case of a system formed by two coupled wave equations, one of them containing a past history term, exponential stabilization depend on the type of coupling mechanism. There are models of this kind whose solutions are not exponentially stable [1,4], in other cases exponential stability is achieved [18,21]. Therefore, the analysis of exponential stability for models consisting of two coupled wave equations, one of them with past history is a subject of great importance and this is the objective of this paper.…”