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, $$u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0,$$ where $\varepsilon>0$ and $p>1$ are constants, driven by the action of a family of double-well potentials of the form $$F(u)=\frac{1}{2n} |1-u^2|^{n},$$ indexed by $n>1$, $n\in\mathbb{R}$ with minima at two pure phases $u = \pm 1$. 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 $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$, persist for an exponentially long time in the critical case with $n=p$, and for an algebraically long time in the supercritical (or degenerate) case with $n > p$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are established. In contrast, in the subcritical case with $n<p$, the transition layer solutions are stationary.