Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations

Pierre Collet, Jean-Pierre Eckmann

We prove that if the initial condition of the Swift-Hohenberg equation ${\partial }_{t}u(x,t)=({\epsilon }^2-(1+{\partial }_x^2{)}^2)u(x,t)-u^3(x,t)$ is bounded in modulus by $C{e}^{-\beta x}$…