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

Abdelkader Braik, Abderrahmane Beniani, Yamina Miloudi

We consider the nonlinear heat equation in a bounded domain with a time varying delay term $$u_{t}+\Delta ^{2}u - J(t)ıt^{t}_{0} g(t-s)\Delta ^{2}u (s)ds+ lpha K(t)u+ \beta K(t)ueft( t-au(t) \right) = 0, $$ 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.