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

Abstract

This paper establishes the emergence of slowly moving transition layer solutions for the p-Laplacian (nonlinear) evolution equation,where ε > 0 and p ≥ 2 are constants, driven by the action of a family of double-well potentials of the formindexed by n ∈ N 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 ε, persist for an exponentially long time in the critical case with 2n = p, and for an algebraically long time in the supercritical (or degenerate) case with 2n > 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 2n < p, the transition layer solutions are stationary.