On trajectory and global attractors for semilinear heat equations with fading memory

In this paper, we construct connected trajectory and global attractors for heat equations with linear fading memory and with nonlinear heat sources. No restriction on the polynomial growth of the nonlinear term is assumed. We also prove the existence of a global Lyapunov function for these equations under proper assumptions on the rate of exponential decay of the memory kernel. The existence of such a Lyapunov function implies that the trajectory and global attractors of the equation under consideration have a regular structure, i.e., they coincide with unstable trajectory sets issuing from the set of stationary points of the equation.