Generation and metastability of patterns for nonlinear evolution equations

Of concern are the properties of solutions of one space dimensional evolution equation utx,t=A u˙,t x,x∈ R,t>0, 0.1 where $A$ is a nonlinear operator which is independent of the time t, maps functions of space variable · to functions of x. Examples of this include some important models such as Allen-Cahn equation, Neural network model and Ising model etc. We show that under certain assumptions on $A$, the solution of the rescaled version of (0.1) utx,t=A ue˙, t xe ,e>0small, will develop a “transition layer structure”, i.e. a pattern, at a predictable time and that this pattern will last for a very long time but will be eventually destroyed.