ON THE SWIFT-HOHENBERG EQUATION WITH SLOW AND FAST DYNAMICS: WELL-POSEDNESS AND LONG-TIME BEHAVIOR

We propose a mathematical analysis of the Swift-Hohenberg equation arising from the phase field theory to model the transition from an unstable to a (meta)stable state. We also consider a recent generalization of the original equation, obtained by introducing an inertial term, to predict fast degrees of freedom in the system. We formulate and prove well-posedness results of the concerned models. Afterwards, we analyse the long-time behavior in terms of global and exponential attractors. Finally, by reading the inertial term as a singular perturbation of the Swift-Hohenberg equation, we construct a family of exponential attractors which is Holder continuous with respect to the perturbative parameter of the system.