EXPONENTIAL DECAY AND BLOW-UP FOR NONLINEAR HEAT EQUATIONS WITH VISCOELASTIC TERMS AND ROBIN-DIRICHLET CONDITIONS

In this article, we consider a system of nonlinear heat equations with viscoelastic terms and Robin-Dirichlet conditions. First, we prove existence and uniqueness of a weak solution. Next, we prove a blow up result of weak solutions with negative initial energy. Also, we give a sufficient condition that guarantees the existence and exponential decay of global weak solutions. The main tools are the Faedo-Galerkin method, a Lyapunov functional, and a suitable energy functional.