On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

Abstract

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295–303) to the nonlinear damping case under weaker assumption on the relaxation function g ( t ) . Then, we give an exponential decay result without the relation between g ′ ( t ) and g ( t ) for the linear damping case, provided that ‖ g ‖ L 1 ( 0 , ∞ ) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.