Global existence, blow-up and optimal decay for a nonlinear viscoelastic equation with nonlinear damping and source term

Abstract

In this paper, we are concerned with a viscoelastic wave equation with memory term, nonlinear damping and source term. Firstly, using the potential well method combined with Galerkin approximation procedure, the global weak solutions are obtained. Secondly, we investigate the blow-up of solutions with initial positive and negative energy, as well as our result improves the earlier ones in [29] and [36]. Finally, under some assumptions imposed on damping coefficient and the relaxation function, we establish the optimal decay of the solutions which conducted by perturbed energy method. Moreover, we obtain that the exponential form of relaxation function lead to better decay result and memory term can slow down the energy decay by displaying the energy decay graphically.