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查看全文