Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations

Abstract

We prove that if the initial condition of the Swift-Hohenberg equation $\partial_t u(x,t)=\bigl(ε^2-(1+\partial_ x^2)^2\bigr) u(x,t) -u^3(x,t)$ is bounded in modulus by $Ce^{-βx}$ as $x\to+\infty $, the solution cannot propagate to the right with a speed greater than $\sup_{0<γ\leβ}γ^{-1}(ε^2+4γ^2+8γ^4).$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift-Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg-Landau equation, where the minimal speed is not determined by the linearization alone.