00-222 Pierre Collet, Jean-Pierre Eckmann
Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations (160K, postscript, 16pp) May 12, 00
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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.

Files: 00-222.src( 00-222.keywords , export.ps )