Reaction-diffusion with memory in the minimal state framework

We consider the integrodifferential equation \[ ∂ _t u-Δ u -∫ _0^∞ κ (s)Δ u(t-s) d s + φ (u)=f\] arising in the Coleman-Gurtin theory of heat conduction with hereditary memory. Within a novel abstract framework, based on the notion of minimal state, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions.