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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 $g'(t)=-\mu u_x(t, g(t)),\ h'(t)=-\mu u_x(t,h(t))$, 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.

Submission history

From: Yihong Du Prof [view email]
[v1] Sun, 14 Jun 2026 07:55:00 UTC (34 KB)