Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: equal limits at infinity

We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution has finite limits $\theta^\pm$ as $x\to\pm\infty$, our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as $t\to\infty$ are steady states. It is known that this result is not valid in general if the limits are equal: $\theta^\pm=\theta_0$. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either $f(\theta_0)\ne0$, or $f(\theta_0)=0$ and $\theta_0$ is a stable equilibrium of the equation $\dot \xi=f(\xi)$. If $f(\theta_0)=0$ and $\theta_0$ is an unstable equilibrium of the equation $\dot \xi=f(\xi)$, we also prove some quasiconvergence theorem making (necessarily) additional assumptions on $u_0$. A major ingredient of our proofs of the quasiconvergence theorems---and a result of independent interest---is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.