Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms

Nadia Mezouar,Salah Mahmoud Boulaaras,Ali Allahem

doi:10.1155/2020/7105387

Abstract

In this paper, a nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay term and logarithmic nonlinearity in the weakly nonlinear internal feedback is considered, where the global and local existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo-Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. These results extend and improve many results in the literature.

Highlights

Bartkowski and Gorka [22] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for equation (6)
We investigate the stabilization of a dynamic model describing a string with a rigid surface and an interior somehow permissive to slight deformations. is leads to a varying material density |ut|l and a Kirchhoff term M(‖∇u‖2) that depends on ‖∇u‖2
We prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo-Galerkin procedure

Summary

Introduction

We consider the global existence and decay properties of solutions for the initial boundary value problem of the following viscoelastic nondegenerate Kirchhoff equation of the form:. Wu [6] treated problem (1) for a constant time delay τ and g1(x) g2(x) x He proved the local existence result using the Faedo-Galerkin method and established the decay result employing suitable Lyapunov functionals under appropriate conditions on μ1 and μ2 and on the kernel h. Ey proved the global existence and uniform decay for the following problem:. Bartkowski and Gorka [22] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for equation (6). H1 is strictly decreasing and convex on (0, 1] with limt⟶0H1(t) +∞

Preliminaries

Proof of Theorem 1

Uniform Decay of the Energy Proof of Theorem 2