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This paper is concerned with the asymptotic behavior of bounded solutions ofthe 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 boundedfunction with compact support and $f$ is a rather general nonlinearity that isperiodic 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 beenestablished by Du and Matano (2010). However, the presence of periodic forcingmakes the problem significantly more difficult, partly because the structure oftime periodic solutions is much less understood than that of steady states. Inthis paper, we first prove that any $\omega$-limit solution is either spatiallyconstant or symmetrically decreasing, even if the initial data is notsymmetric. Furthermore, we show that the set of $\omega$-limit solutions eitherconsists of a single time-periodic solution or it consists of multipletime-periodic solutions and heteroclinic connections among them. Next, under amild non-degenerate assumption on the corresponding ODE, we prove that the$\omega$-limit set is a singleton, which implies the solution converges to atime-periodic solution. Lastly, we apply these results to equations withbistable nonlinearity and combustion nonlinearity, and specify more preciselywhich time-periodic solutions can possibly be selected as the limit.

Dynamics of Time-Periodic Reaction-Diffusion Equations with Compact Initial Support on R