Large time behavior of ODE type solutions to nonlinear diffusion equations

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear diffusion PDE, showing they resemble solutions to an associated ODE and tend to infinity, with detailed descriptions of their asymptotic behavior.

Contribution

It provides a precise characterization of the large time asymptotics of solutions to a nonlinear diffusion equation, linking their behavior to the underlying diffusion effects.

Findings

Solutions behave like ODE solutions as time goes to infinity

Solutions tend to infinity over time

The asymptotic behavior depends on the nonlinear diffusion effects

Abstract

Consider the Cauchy problem for a nonlinear diffusion equation \begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=\Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0)=\lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} where $m>0$, $\alpha\in(-\infty,1)$, $\lambda>0$ and $\varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N)$ with $1\le r<\infty$ and $\inf_{x\in{\bf R}^N}\varphi(x)>-\lambda$. Then the positive solution to problem (P) behaves like a positive solution to ODE $\zeta'=\zeta^\alpha$ in $(0,\infty)$ and it tends to $+\infty$ as $t\to\infty$. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.