Pattern selection of solutions of the Swift–Hohenberg equation

We study the large-time behaviour of solutions u(x,t) of the Swift–Hohenberg equation on a one-dimensional domain (0,L), focusing in particular on the role of the eigenvalue parameter α and the length L of the domain on the selection of the limiting profiles v. We show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles, and prove the existence of a collection Σn of disjoint intervals, which depend on the value of α, such that if L∈Σn then solutions all converge to the trivial solution v=0. We show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches. We establish global bounds for the solutions u(x,t) and identify the global attractor. Finally, we derive an estimate about the shape of the final state.