A Rigorous Upper Bound on the Propagation Speed for the Swift–Hohenberg and Related Equations

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Abstract

We prove that if the initial condition of the Swift–Hohenberg equation

is bounded in modulus by Ce−βx as x→+∞, the solution cannot propagate to the right with a speed greater than

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 critical speed is not determined by the linearization alone.

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