Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions

TL;DR

This paper analyzes the long-term behavior of solutions to a reaction-diffusion equation with free boundaries, revealing conditions under which solutions vanish, shrink, or stabilize, especially for cases where the boundary expansion rate is positive.

Contribution

It extends previous work by examining the case lpha>0, showing that shrinking can occur and is linked to vanishing, and provides a comprehensive analysis of asymptotic behaviors for various nonlinearities.

Findings

Shrinking occurs when boundary width tends to zero, linked to solution vanishing.

Bounded positive solutions tend to a nonzero stationary state as time approaches infinity.

Complete characterization of solution behaviors for monostable and bistable nonlinearities.

Abstract

We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+\alpha$ and $h'(t)=-u_x(t,g(t))-\alpha$ for some $\alpha>0$. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When $\alpha=0$, the problem was recently investigated by \cite{DuLin, DuLou}. In this paper we consider the case $\alpha>0$. In this case shrinking (i.e. $h(t)-g(t)\to 0$) may happen, which is quite different from the case $\alpha=0$. Moreover, we show that, under certain conditions on $f$, shrinking is equivalent to vanishing (i.e. $u\to 0$), both of them happen as $t$ tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as $t\to \infty$. As applications, we consider monostable and bistable types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.