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) = integral(t)(0) f (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 u(q) to the equation satisfy vertical bar vertical bar u(q)vertical bar vertical bar(infinity) -> 0, as q down arrow 0. (C) 2017 Elsevier Inc. All rights reserved.