We consider  the semilinear parabolicequation $u_t=\Delta u+f(u)$ on $R^N$,
assuming that  $f$ is an arbitrary $C^1$ function satisfying $f(0)=0$ and
$f'(0) < 0$. We prove that any bounded positive solution that decays to zero at
spatial infinity, uniformly with respect to $t$, converges to a (single) stationary
solution as $t\to\infty$. Our proof combines energy and comparison techniques
with dynamical system arguments. We first establish  an asymptotic symmetrization
result: as $t\to \infty$, $u(x,t)$ approaches a set of of steady states that are radially
symmetric about a common origin in $R^N$. To this aim we introduce a new tool
that we call  first moments of energy. Having established the symmetrization, we
apply a general convergence result for gradient-like dynamical  systems. This
amounts to showing that  the dimension of the kernel of the linearized operator
around an equilibrium $w$ matches the dimension of  a manifold of  equilibria
passing  through $w$.