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

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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.

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