 00222 Pierre Collet, JeanPierre Eckmann
 Proof of the marginal stability bound for the
SwiftHohenberg equation and related equations
(160K, postscript, 16pp)
May 12, 00

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Abstract. We prove that if the initial condition of the
SwiftHohenberg 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 longstanding conjecture about the possible asymptotic
propagation speed of the SwiftHohenberg 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 GinzburgLandau equation, where the minimal speed is not
determined by the linearization alone.
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