A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior

We study a free boundary problem of a reaction-diffusion equation \begin{document}$ u_t = \Delta u+f(u) $\end{document} for \begin{document}$ t>0,\ |x| under a radially symmetric environment in \begin{document}$ \mathbb{R}^N $\end{document}. The reaction term \begin{document}$ f $\end{document} has positive bistable nonlinearity, which satisfies \begin{document}$ f(0) = 0 $\end{document} and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface \begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}, which expands to infinity as \begin{document}$ t\to\infty $\end{document}, even when the corresponding semi-wave problem does not admit solutions.