  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.
