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

Yuki Kaneko; Hiroshi Matsuzawa; Yoshio Yamada

doi:10.3934/dcds.2021209

Article Contents

2022, Volume 42, Issue 6: 2719-2745. Doi: 10.3934/dcds.2021209

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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

1.
Department of Mathematical and Physical Sciences, Japan Women's University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681, Japan
2.
Department of Mathematics and Physics, Faculty of Science, Kanagawa University, Tsuchiya 2946, Hiratsuka-city, Kanagawa 259-1293, Japan
3.
Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

^* Corresponding author: Yuki Kaneko
^* Corresponding author: Yuki Kaneko

Received: February 2021

Revised: November 2021

Early access: January 2022

Published: June 2022

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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Abstract

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.

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

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References

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