Finite propagation and saturation in reaction-diffusion-advection equations governed by p-Laplacian operator

TL;DR

This paper investigates the behavior of traveling wave solutions in a reaction-diffusion-advection equation involving a p-Laplacian operator, providing criteria for their finite-time saturation, continuability, and classification.

Contribution

It offers new criteria to classify traveling wave solutions, including conditions for finite-time saturation and regularity, in a complex reaction-diffusion-advection model with p-Laplacian.

Findings

Criteria for finite-time attainment of equilibria

Conditions for solution continuability as $C^1$-solutions

Classification of solutions as sharp or smooth

Abstract

The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation \[f(u)u_x + g(u)u_\tau = [d(u)|u_x|^{p-2} u_x]_x+ \rho(u), \quad (x,\tau)\in \R\times [0,+\infty).\] Besides existence and non-existence results for traveling wave solutions, the main focus is their classification: we provide criteria to establish if they attain one or both the equilibria at a finite time and in this case, if they are continuable as $C^1$-solutions or if they are sharp solutions.