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, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] 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.

Figures (5)

• Figure 1: Plots of the potential function \ref{['defF']} for $n={2,4,6}$, which underly different behaviors when compared to the diffusion parameter $p \geq 2$. For example, when $p = 4$ the former cases correspond to subcritical ($n < p$), critical ($n=p$) and supercritical or degenerate ($n>p$) cases, respectively.
• Figure 2: The level of the energy (given by the constant $C$) has to be such that $C> -F(u)$. In particular, bounded nontrivial solutions can be found only for $C \in (-1/2n, 0)$. When $C=0$ we have the heteroclinic solution (that touches the value $\pm 1$ only in the case $n<p$).
• Figure 3: Solutions to \ref{['eq:P-model']} for $\varepsilon=0.1$, $p=n=2$ (left), and $p=n=4$ (right); the initial datum $u_0$ is as in \ref{['eq:translayer']} with $6$ transition points located at $(-3.4,-2,-0.5, 0.8, 2.2, 3.2).$
• Figure 4: Solutions to \ref{['eq:P-model']} for $\varepsilon=0.1$, $p=2$ (left) and $p=3$ (right); the initial datum $u_0$ is as in Figure \ref{['fig:2n=p']}, while the potential $F$ is as in \ref{['defF2']} with $n=4$.
• Figure 5: Solutions to \ref{['eq:P-model']} for $\varepsilon=0.1$, $n=8$ and $p=\pi$ (left) and $p=5.5$ (right); the initial datum $u_0$ is as in Figure \ref{['fig:2n=p']}.

Theorems & Definitions (34)

• Remark 1.1
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5
• proof
• Proposition 2.6