Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III:unstable limit at infinity

TL;DR

This paper studies the long-term behavior of solutions to a semilinear parabolic equation on the real line, focusing on cases where the equilibrium at infinity is unstable, and proves near-optimal quasiconvergence results under minimal conditions.

Contribution

It extends previous work by removing technical restrictions and establishing quasiconvergence for solutions approaching an unstable equilibrium at infinity, assuming only nonoscillatory behavior.

Findings

Proves quasiconvergence for solutions with unstable limits at infinity.

Removes restrictive technical conditions from earlier results.

Shows solutions are nearly always quasiconvergent under minimal assumptions.

Abstract

This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $\theta^\pm$ as $x\to\pm\infty$. We assume that $f$ is a locally Lipschitz function on $\mathbb{R}$ satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In the first two parts of this series we mainly considered the cases where either $\theta^-\neq \theta^+$; or $\theta^\pm=\theta_0$ and $f(\theta_0)\ne0$; or else $\theta^\pm=\theta_0$, $f(\theta_0)=0$, and $\theta_0$ is a stable equilibrium of the equation $\dot \xi=f(\xi)$. In all these cases we proved that the corresponding solution $u$ is quasiconvergent -- if bounded -- which is to say that all limit profiles of $u(\cdot,t)$ as $t\to\infty$ are steady states. The limit profiles, or accumulation points, are taken in $L^\infty_{loc}(\mathbb{R})$. In the present paper, we take on the case that $\theta^\pm=\theta_0$, $f(\theta_0)=0$, and $\theta_0$ is an unstable equilibrium of the equation $\dot \xi=f(\xi)$. Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $u(\cdot,t)$ is that it is nonoscillatory (has only finitely many critical points) at some $t\geq 0$. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.