Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction

TL;DR

This paper analyzes the long-term behavior of solutions to a degenerate diffusion equation with a mixed monostable-bistable reaction term, classifying possible asymptotic states and constructing traveling wave solutions with free boundaries.

Contribution

It provides a trichotomy classification of asymptotic behaviors and constructs sharp traveling wave solutions for the generalized degenerate diffusion equation.

Findings

Solutions exhibit small-spreading, transition, or big-spreading behavior as time goes to infinity.

Constructed classical and sharp traveling wave solutions with free boundaries.

Characterized the spreading solutions near their fronts using these traveling waves.

Abstract

We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u)]_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A'(0)=0$ and $A(u),\ A'(u),\ A''(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$ and of bistable type in $[s_1,1]$. We first give a trichotomy result on the asymptotic behavior of the solutions starting at compactly supported initial data, which says that, as $t\to \infty$, either small-spreading (which means $u$ tends to $s_1$), or transition, or big-spreading (which means $u$ tends to $1$) happens for a solution. Then we construct the classical and sharp traveling waves (a sharp wave means a wave having a free boundary which satisfies the Darcy's law) for the generalized degenerate diffusion equation, and then using them to characterize the spreading solution near its front.