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.