On Classical Solutions for a Swift-Hohenberg Type Equation

The Swift-Hohenberg equation models the evolution of the hydrodynamic fluctuations at the convective instability. It describes also the mechanism of shear microbands formation in nanocrystalline materials, the amplitude of optical electric field inside the cavity, the patterns inside thin vibrated granular layers. It is also a model of the population dynamics, or a model for ultrafast pulse propagation in a mode-locked laser cavity in the few-femtosecond pulse regime. Under appropriate assumptions of the coefficients of such equation, in this paper, we prove the well-posedness of the classical solutions of the Cauchy problem.