Global Existence and Decay of Solutions for Coupled Nondegenerate Kirchhoff System with a Time Varying Delay Term

Nadia Mezouar,Sultan Alodhaibi,Salem Alkhalaf,Salah Mahmoud Boulaaras

doi:10.1155/2020/6324971

Abstract

This paper deals with the global existence of solutions in a bounded domain for nonlinear viscoelastic Kirchhoff system with a time varying delay by using the energy and Faedo–Galerkin method with respect to the delay term weight condition in the feedback and the delay speed. Furthermore, by using some convex functions properties, we prove a uniform stability estimate.

Highlights

In [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation:
Under assumption setting on g1, g2, σ, and τ, the authors have obtained the global existence of solution and the decay rate of energy
We introduce, as in [8], the new variables z1(x, ρ, t) ut(x, t − ρτ(t)), z2(x, ρ, t) vt(x, t − ρτ(t)), x ∈ Ω, ρ ∈ (0, 1), t > 0, x ∈ Ω, ρ ∈ (0, 1), t > 0

Summary

Introduction

In [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation: Under assumption setting on g1, g2, σ, and τ, the authors have obtained the global existence of solution and the decay rate of energy. We can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Faedo–Galerkin procedure, and under a choice of a suitable Lyapunov functional, we establish an exponential decay result.

Results

Conclusion