Existence, general decay and blow-up of solutions for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and dynamic boundary conditions

TL;DR

This paper investigates the existence, decay, and blow-up of solutions for a nonlinear viscoelastic Kirchhoff equation with complex damping and boundary conditions, providing conditions for global existence and finite-time blow-up.

Contribution

It establishes local existence, global existence under stable initial data, blow-up under unstable data, and a general decay rate for the energy, extending previous results.

Findings

Solutions exist locally via Faedo-Galerkin method.

Global solutions occur for stable initial data.

Solutions blow up in finite time for unstable initial data.

Abstract

Our aim in this article is to study a nonlinear viscoelastic Kirchhoff equation with strong damping, Balakrishnan-Taylor damping, nonlinear source and dynamical boundary condition. Firstly, we prove the local existence of solutions by using the Faedo-Galerkin approximation method combined with a contraction mapping theorem. We then prove that if the initial data enter into the stable set, the solution globally exists, and if the initial data enter into the unstable set, the solution blows up in a finite time. Moreover, we obtain a general decay result of the energy, from which the usual exponential and polynomial decay rates are only special cases.