  This paper is concerned with the integrodifferential equation $$\partial_t
u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f$$ arising
in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in
presence of a nonlinearity $\varphi$ 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.
