Development of interfaces in ℝ N

Consider the reaction-diffusion equation in ℝ N × ℝ + : u t − h 2 Δ u + Φ( u ) = 0, where Φ is the derivative of a bistable even potential, and h is a small parameter. If the initial data have a smooth noncritical zero set, we prove that an interface appears in time O (log ( h −1 )), and that the solution stays close to it for at least time O (1/√ h ).