On a class of nonlinear heat equations with viscoelastic term

Abstract

The main goal of this paper is to study a model of the strongly nonlinear heat equation with viscoelastic term and nonlinear interior source of the form 1 + a|u

q−2 ut − ∆u + R t 0 g(t − s)∆u(s)ds = f(u), in Ω × [0, ∞), u = 0 on ∂Ω × [0, ∞), u(x, 0) = u0(x) in Ω.We show the results of local (or global) existence of weak solutions by using the Galerkin approximation method. In addition it has been provided sufficient conditions for the large time decay and the finite time blow-up of the weak solutions.