We consider bounded solutions of the Cauchy problem
\[
\begin{aligned}
    u_t-\Delta u=f(u),\;\; &x\in \Bbb R^N, \;t>0,\\
 u(0,x)=u_0(x),\;\; &   x\in\Bbb R^N,
  \end{aligned}
  \]
where $u_0$ is a nonnegative function with compact support and $f$ is a $C^1$ function on $\Bbb R$ with $f(0)=0$. Assuming a minor nondegeneracy condition on $f$, we prove that, as $t\to\infty$, the solution $u(\cdot,t)$ converges to an equilibrium $\varphi$ locally uniformly in $\Bbb R^N$.  Moreover, the limit function $\varphi$ is either a constant equilibrium,  or there is a point $x_0\in\Bbb R^N$ such that $\varphi$ is radially symmetric and radially decreasing about $x_0$, and it approaches a constant equilibrium as $|x-x_0|\to\infty$. The  nondegeneracy condition only concerns a specific set of  zeros of $f$ and we make no assumption whatsoever on the nonconstant equilibria. The set of such equilibria can be very complicated and  indeed a complete understanding of this set is usually beyond reach in dimension $N\geq2$. Moreover, due to the symmetries of the equation, there are always continua of such equilibria. Our result shows that the assumption ``$u_0$ has compact support'' is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the $\omega$-limit set of $u$ have a rather simple structure and, second, exactly one of them is selected. Our convergence result remains valid if $\Delta u$ is replaced by a general elliptic operator of the form $\sum_{i,j} a_{ij}u_{x_ix_j}$ with constant coefficients $a_{ij}$.