  We consider the Dirichlet problem u_t &= \Delta u + f(x, u, \nabla u)+ h(x,
t),& \qquad &(x, t) \in \Omega \times (0, \infty), u &= 0, & \qquad &(x, t) \in
\partial\Omega \times (0, \infty), on a bounded domain $\Omega \subset
\mathbb{R}^N$. The domain and the nonlinearity $f$ are assumed to be invariant
under the reflection about the $x_1$-axis, and the function $h$ accounts for a
nonsymmetric decaying perturbation: $h(\cdot, t)\to 0$ as $t\to\infty$. In one
of our main theorems, we prove the asymptotic symmetry of each bounded positive
solution $u$. The novelty of this result is that the asymptotic symmetry is
established even for solutions that are not assumed uniformly positive. In
particular, some equilibria of the limit time-autonomous problem (the problem
with $h\equiv 0$) with a nontrivial nodal set may occur in the $\omega$-limit
set of $u$ and this prevents one from applying common techniques based on the
method of moving hyperplanes. The goal of our second main theorem is to
classify the positive entire solutions of the time-autonomous problem. We prove
that if $U$ is a positive entire solution, then one of the following applies:
(i) for each $t\in \mathbb{R}$, $U(\cdot,t)$ is even in $x_1$ and decreasing in
$x_1>0$, (ii) $U$ is an equilibrium, (iii) $U$ is a connecting orbit from an
equilibrium with a nontrivial nodal set to a set consisting of functions which
are even in $x_1$ and decreasing in $x_1>0$, (iv) is a heteroclinic connecting
orbit between two equilibria with a nontrivial nodal set.
