Convergence of Solutions of the Porous Medium Equation with Reactions

TL;DR

This paper proves that solutions to the one-dimensional porous medium equation with reactions converge over time to stationary states, providing a complete classification of long-term behaviors for various nonlinearities.

Contribution

It establishes a general convergence theorem and classifies asymptotic behaviors for PME with monostable, bistable, and combustion nonlinearities.

Findings

Solutions converge to stationary states or zeros of the reaction term.

Complete classification of asymptotic behaviors for different nonlinearities.

Convergence holds for bounded solutions with nonnegative, compactly supported initial data.

Abstract

Consider the Cauchy problem of one dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as $t\to \infty$ to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behaviors of the solutions for PME with monostable, bistable and combustion types of nonlinearities.