Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source

Abstract

This paper deals with the initial-boundary value problem for the viscoelastic equation with strong damping and nonlinear source. Firstly, we prove the local existence of solutions by using the Faedo-Galerkin approximation method and Contraction Mapping Theorem. By virtue of the potential well theory and convexity technique, we then prove that if the initial data enter into the stable set, then the solution globally exists and decays to zero with a polynomial rate, and if the initial data enter into the unstable set, then the solution blows up in a finite time. Moreover, we show that the solution decays to zero with an exponential or polynomial rate depending on the decay rate of the relaxation function.