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 ${\partial }_{t}u(x,t)=({\epsilon }^2-(1+{\partial }_x^2{)}^2)u(x,t)-u^3(x,t)$ is bounded in modulus by $C{e}^{-\beta x}$ as $x\to +\infty$, the solution cannot propagate to the right with a speed greater than ${\sup }_{0<\gamma \le \beta }{\gamma }^{-1}({\epsilon }^2+4{\gamma }^2+8{\gamma }^4).$ 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.

16 pages, 0 figures