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

  • * Corresponding author: Yuki Kaneko

    * Corresponding author: Yuki Kaneko 

1 Partially supported by Grant-in-Aid for Early-Career Scientists (19K14602).
2 Partially supported by Grant-in-Aid for Scientific Research (C) 17K05340 and 20K03709.
3 Partially supported by Grant-in-Aid for Scientific Research (C) 19K03573

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  • We study a free boundary problem of a reaction-diffusion equation $ u_t = \Delta u+f(u) $ for $ t>0,\ |x|<h(t) $ under a radially symmetric environment in $ \mathbb{R}^N $. The reaction term $ f $ has positive bistable nonlinearity, which satisfies $ f(0) = 0 $ 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 $ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $, which expands to infinity as $ t\to\infty $, even when the corresponding semi-wave problem does not admit solutions.

    Mathematics Subject Classification: Primary: 35R35, 35K57, 35J61, 34B15; Secondary: 92D25.

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