Long time dynamics of solutions to p-laplacian diffusion problems with bistable reaction terms

This paper establishes the emergence of slowly moving transition layer solutions for the p-Laplacian (nonlinear) evolution equation, ut = "p(juxjp-2ux)x - F0(u); x 2 (a; b); t > 0; where " > 0 and p > 1 are constants, driven by the action of a family of double-well potentials of the form F(u) = 1 2θ j1 - u2jθ; indexed by θ > 1, ∈ 2 R with minima at two pure phases u = ±1. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to ±1 except at a finite number of thin transitions of width ", persist for an exponen- tially long time in the critical case with θ = p, and for an algebraically long time in the supercritical (or degenerate) case with θ > p. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg{Landau type are established. In contrast, in the subcritical case with θ < p, the transition layer solutions are stationary.