Pierre Collet, Jean-Pierre Eckmann
Proof of the marginal stability bound for the
Swift-Hohenberg equation and related equations
(160K, postscript, 16pp)
ABSTRACT. We prove that if the initial condition of the
Swift-Hohenberg equation
$$\partial _t u(x,t)=\bigl(\epsilon^2-(1+\partial_ x^2)^2\bigr) u(x,t)
-u^3(x,t)
$$
is bounded in modulus by $Ce^{-\beta x }$ 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.