| Abstract: |
| This work 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, \vspace{3pt}\ 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 periodic in $t$ and satisfies $f(\cdot,0)=0$. We first prove that the $\omega$-limit set of any bounded solution either consists of a single time-periodic solution or it consists of time-periodic solutions as well as heteroclinic solutions connecting them. Furthermore, under a minor nondegenerate assumption on time-periodic solutions of the corresponding ODE, the convergence to a time-periodic solution is proved. 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. |
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