On the regularity of global attractors

This note is focused on a novel technique in order to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-\Delta u_t-\Delta u+\phi(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.