Note on a dispersive-dissipative wave equation with viscoelasticity

Abstract

This paper deals with an initial-boundary value problem of the dispersive-dissipative wave equation having viscoelastic damping term and a nonlinear source. By applying the Faedo–Galerkin method in conjunction with the contraction mapping principle, we rigorously prove the local existence and uniqueness of weak solutions. Within the potential well theory framework, we categorize the initial data into subcritical, critical, and high initial energy levels. This classification enables us to derive optimal conditions for determining whether the solutions exist globally or exhibit finite-time blow-up behavior. By means of Nakao’s lemma and through the refined analysis of calculus inequalities, such as the embedding theorem, an exponential decay estimate of the energy functional is obtained. Additionally, through the innovative construction of auxiliary functionals, we obtain explicit upper and lower bounds for the blow-up time. These bounds provide a quantitative description of how the blow-up dynamics depend on the initial energy levels, thereby offering deeper insights into the long-term behavior of the solutions.