Well-Posedness and General Decay of Solutions for the Heat Equation with a Time Varying Delay Term

Abstract

We consider the nonlinear heat equation in a bounded domain with a time varying delay term ut + ∆2u − J(t) Z t 0 g(t − s)∆2u(s)ds + αK(t)u + βK(t)u (t − τ (t)) = 0, with initial conditions. By introducing suitable energy and Lyapunov functionals, under some assumptions, we then prove a general decay result of the energy associated of this system under some conditions