Existence and asymptotic behavior of nontrivial solutions to the Swift-Hohenberg equation

In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u⁗+qu″+f(u)=0, under various hypotheses on the parameter q and on the potential F(t)=∫0tf(s)ds, generally assumed to be bounded from below. We prove a non-existence result in the case q≤0 and an existence result of periodic solution for: 1) almost every suitably small (depending on F), positive values of q; 2) all suitably large (depending on F) values of q. Finally, we describe some conditions on F which ensure that some (or all) solutions uq to the equation satisfy ‖uq‖∞→0, as q↓0.