Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III:unstable limit at infinity (2112.11160v2)

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.