Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

TL;DR

This paper studies reaction-diffusion equations of porous medium type, analyzing solution propagation, convergence to steady states, and traveling wave behavior, including conditions for convergence and wave speed determination.

Contribution

It characterizes conditions under which solutions converge to 1 or 0, and analyzes the existence and speed of traveling waves in porous medium reaction-diffusion equations.

Findings

Solutions can converge to 1 or 0 depending on reaction terms.

Existence of a unique traveling wave with finite front.

Traveling wave speed determines asymptotic velocity of solutions.

Abstract

We consider reaction-diffusion equations of porous medium type, with different kind of reaction terms, and nonnegative bounded initial data. For all the reaction terms under consideration there are initial data for which the solution converges to 1 uniformly in compact sets for large times. We will characterize for which reaction terms this happens for all nontrivial nonnegative initial data, and for which ones there are also solutions converging uniformly to 0. Problems in this family have a unique (up to translations) travelling wave with a finite front and we will see how its speed gives the asymptotic velocity of all the solutions with compactly supported initial data. We will also prove in the one-dimensional case that solutions with bounded compactly supported initial data converging to 1 do so approaching a translation of this unique traveling wave. We will prove a similar result for non-compactly supported initial data in a certain class.