  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.
