Blow up of Solution for a Nonlinear Viscoelastic Problem with Internal Damping and Logarithmic Source Term

Abstract

This paper is concerned with blow up of weak solutions of the following nonlinear viscoelastic problem with internal damping and logarithmic source term |ut|ρutt + M(∥u∥2)(-∆u) - ∆utt + Z0t g(t - s)∆u(s)ds + ut = u|u|p R-2 ln |u|k R  with Dirichlet boundary initial conditions in a bounded domain Ω ⊂ Rn. In the physical point of view, this is a type of problems that usually arises in viscoelasticity. It has been considered with power source term first by Dafermos [3], in 1970, where the general decay was discussed. We establish conditions of p, ρ and the relaxation function g, for that the solutions blow up in finite time for positive and nonpositive initial energy. We extend the result in [15] where is considered M = 1 and external force type |u|p-2u in it. Further we state and sketch the proof of a result of local existence of weak solution that is used in the proof of the theorem on blow up. The idea underlying the proof of local existence of solution is based on Faedo-Galerkin method combined with the Banach fixed point method.