Semi-wave and sharp estimates of propagation for monostable free boundary problems in time-periodic environment

Abstract

We investigate the propagation profile of positive solutions to \begin{equation*} u_t-du_{xx}=f(t,u) \mbox{ for } t>0,\ x\in(g(t),h(t)), \end{equation*} where $f(t,u)$ is monostable in $u$ and $T$-periodic in $t$, and the free boundaries $x=g(t),x=h(t)$ are determined by the Stefan condition $g^{\prime }(t)=-\mu u_x(t,g(t)),h^{\prime }(t)=-\mu u_x(t,h(t))$, coupled with $u(t,g(t))=u(t,h(t))=0$. For a special nonlinearity satisfying the strong KPP condition, the long-time behavior and asymptotic spreading speed of this problem were considered by Du, Guo and Peng \cite{DGP}. In this paper, by employing new techniques, we extend the results of \cite{DGP} to general monostable nonlinearities beyond the KPP framework and at the same time we obtain more precise description of the propagation profile: we prove the existence and uniqueness of a semi-wave and show that the spreading solution converges to this semi-wave as time goes to infinity.