Dissipative mechanism and global attractor for modified Swift-Hohenberg equation in $R^{N}$

Abstract

A Cauchy problem for a modification of the Swift-Hohenberg equation in $R^{N}$ with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in $H^2(R^{N})$ in particular.