  This paper is concerned with the asymptotic behavior of bounded solutions of
the Cauchy problem \begin{equation*} \left\{ \begin{array}{ll} u_t=u_{xx}
+f(t,u), & x\in\mathbb{R},\,t>0,\\ u(x,0)= u_0, & x\in\mathbb{R},
\end{array}\right. \end{equation*} where $u_0$ is a nonnegative bounded
function with compact support and $f$ is a rather general nonlinearity that is
periodic in $t$ and satisfies $f(\cdot,0)=0$. In the autonomous case where
$f=f(u)$, the convergence of every bounded solution to an equilibrium has been
established by Du and Matano (2010). However, the presence of periodic forcing
makes the problem significantly more difficult, partly because the structure of
time periodic solutions is much less understood than that of steady states. In
this paper, we first prove that any $\omega$-limit solution is either spatially
constant or symmetrically decreasing, even if the initial data is not
symmetric. Furthermore, we show that the set of $\omega$-limit solutions either
consists of a single time-periodic solution or it consists of multiple
time-periodic solutions and heteroclinic connections among them. Next, under a
mild non-degenerate assumption on the corresponding ODE, we prove that the
$\omega$-limit set is a singleton, which implies the solution converges to a
time-periodic solution. Lastly, we apply these results to equations with
bistable nonlinearity and combustion nonlinearity, and specify more precisely
which time-periodic solutions can possibly be selected as the limit.
