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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 whoseinitial 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 describethe asymptotic behavior of $u(x,t)$ as $t\to\infty$. In the first two parts ofthis 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 equilibriumof the equation $\dot \xi=f(\xi)$. In all these cases we proved that thecorresponding solution $u$ is quasiconvergent -- if bounded -- which is to saythat all limit profiles of $u(\cdot,t)$ as $t\to\infty$ are steady states. Thelimit 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 unstableequilibrium of the equation $\dot \xi=f(\xi)$. Our earlier quasiconvergencetheorem in this case involved some restrictive technical conditions on thesolution, which we now remove. Our sole condition on $u(\cdot,t)$ is that it isnonoscillatory (has only finitely many critical points) at some $t\geq 0$.Since it is known that oscillatory bounded solutions are not alwaysquasiconvergent, our result is nearly optimal.

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