Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time

Abstract

In this study, we investigate a nonlinear heat equation incorporating both a viscoelastic term and a reaction-diffusion term that depends on space-time variables. Initially, we establish the local Hadamard well-posedness results using the standard Faedo-Galerkin method. Subsequently, we demonstrate that the solution exhibits finite-time blowup for initial energy values that are both negative and nonnegative. Finally, we establish the global existence of the solution and provide general decay estimates for the energy functions with small initial energy, utilizing Martinez’s inequality.