Proof of the Marginal Stability Bound for the Swift-Hohenberg Equation and Related Equations

We prove that if the initial condition of the Swift-Hohenberg equation tu(x; t) = "2 (1 + 2 x)2 u(x; t) u3(x; t) is bounded in modulus by Ce x as x ! +1, the solution cannot propagate to the right with a speed greater than sup 0< 1("2 + 4 2 + 8 4) : This settles a long-standing 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 Ginzburg-Landau equation, where the minimal speed is not determined by the linearization alone.