Spreading and vanishing in nonlinear diffusion problems with free boundaries

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear diffusion problems with free boundaries, revealing conditions for spreading or vanishing and characterizing the asymptotic speed of the expanding front.

Contribution

It extends the analysis of free boundary problems to general nonlinearities, providing a comprehensive description of the dynamics and thresholds for spreading and vanishing.

Findings

Existence of a threshold parameter determining spreading or vanishing.

Complete characterization of long-term behavior for monostable, bistable, and combustion nonlinearities.

Determination of asymptotic spreading speed using semi-waves.

Abstract

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [8]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $\omega(u)$ of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter $\sigma$ in the initial data, we reveal a threshold value $\sigma^*$ such that spreading ($\lim_{t\to\infty}u= 1$) happens when $\sigma>\sigma^*$, vanishing ($\lim_{t\to\infty}u=0$) happens when $\sigma<\sigma^*$, and at the threshold value $\sigma^*$, $\omega(u)$ is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.