Asymptotics of the Coleman-Gurtin model

This paper is concerned with the integrodifferential equation $${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u-{\int }_0^{\mathrm{\infty }}\kappa (s)\mathrm{\Delta }u(t-s)\text{d}s+\phi (u)=f$$ arising in the Coleman-Gurtin’s theory of heat conduction with hereditary memory, in presence of a nonlinearity $\phi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.