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

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 $θ^\pm$ 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 $θ^-\neq θ^+$; or $θ^\pm=θ_0$ and $f(θ_0)\ne0$; or else $θ^\pm=θ_0$, $f(θ_0)=0$, and $θ_0$ is a stable equilibrium of the equation $\dot ξ=f(ξ)$. 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 $θ^\pm=θ_0$, $f(θ_0)=0$, and $θ_0$ is an unstable equilibrium of the equation $\dot ξ=f(ξ)$. 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.