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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

Received  February 2021 Revised  November 2021 Early access January 2022

Fund Project: 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

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.

Citation: Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021209
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, pp. 5–49, Lecture Notes in Math., 446, Springer, Berlin, 1975.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [3] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157.  doi: 10.1512/iumj.1981.30.30012.  Google Scholar [4] G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar [5] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.  Google Scholar [6] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955.  Google Scholar [7] E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2002), 1891-1899.  doi: 10.1090/S0002-9939-02-06733-3.  Google Scholar [8] Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [9] Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar [10] Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar [11] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405, SIAM J. Math. Anal., 45 (2013), 1995–1996 (Erratum). doi: 10.1137/110822608.  Google Scholar [12] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar [13] Y. Du, B. Lou and M. Zhou, Spreading and vanishing for nonlinear Stefan problems in high space dimensions, J. Elliptic Parabol. Equ., 2 (2016), 297-321.  doi: 10.1007/BF03377406.  Google Scholar [14] Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Rational Mech. Anal., 212 (2014), 957-1010.  doi: 10.1007/s00205-013-0710-0.  Google Scholar [15] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar [16] Y. Du, H. Matsuzawa and M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008.  Google Scholar [17] M. Endo, Y. Kaneko and Y. Yamada, Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity, Discrete and Contin. Dyn. Syst., Series A, 40 (2020), 3375-3394.  doi: 10.3934/dcds.2020033.  Google Scholar [18] Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal. Real World Appl., 18 (2014), 121-140.  doi: 10.1016/j.nonrwa.2014.01.008.  Google Scholar [19] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of reaction-diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970.  Google Scholar [20] Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅱ : Asymptotic profiles of solutions and radial terrace solution, preprint. Google Scholar [21] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar [22] Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar [23] C. Lei and Y. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar [24] X. Liu and B. Lou, On a reaction-diffusion equation with Robin and free boundary conditions, J. Differential Equations, 259 (2015), 423-453.  doi: 10.1016/j.jde.2015.02.012.  Google Scholar [25] D. Ludwig, D. G. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258.  doi: 10.1007/BF00276310.  Google Scholar [26] L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in $\mathbb{R}^N$, J. Differential Equations, 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar [27] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar [28] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorem for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.  Google Scholar [29] N. Shigesada and K. Kawasaki, Biological Invasions; Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. Google Scholar [30] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

show all references

##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, pp. 5–49, Lecture Notes in Math., 446, Springer, Berlin, 1975.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [3] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157.  doi: 10.1512/iumj.1981.30.30012.  Google Scholar [4] G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar [5] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.  Google Scholar [6] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955.  Google Scholar [7] E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2002), 1891-1899.  doi: 10.1090/S0002-9939-02-06733-3.  Google Scholar [8] Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [9] Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar [10] Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar [11] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405, SIAM J. Math. Anal., 45 (2013), 1995–1996 (Erratum). doi: 10.1137/110822608.  Google Scholar [12] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar [13] Y. Du, B. Lou and M. Zhou, Spreading and vanishing for nonlinear Stefan problems in high space dimensions, J. Elliptic Parabol. Equ., 2 (2016), 297-321.  doi: 10.1007/BF03377406.  Google Scholar [14] Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Rational Mech. Anal., 212 (2014), 957-1010.  doi: 10.1007/s00205-013-0710-0.  Google Scholar [15] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.  Google Scholar [16] Y. Du, H. Matsuzawa and M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008.  Google Scholar [17] M. Endo, Y. Kaneko and Y. Yamada, Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity, Discrete and Contin. Dyn. Syst., Series A, 40 (2020), 3375-3394.  doi: 10.3934/dcds.2020033.  Google Scholar [18] Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal. Real World Appl., 18 (2014), 121-140.  doi: 10.1016/j.nonrwa.2014.01.008.  Google Scholar [19] Y. Kaneko, H. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of reaction-diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970.  Google Scholar [20] Y. Kaneko, H. Matsuzawa and Y. Yamada, A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions Ⅱ : Asymptotic profiles of solutions and radial terrace solution, preprint. Google Scholar [21] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar [22] Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar [23] C. Lei and Y. Du, Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 895-911.  doi: 10.3934/dcdsb.2017045.  Google Scholar [24] X. Liu and B. Lou, On a reaction-diffusion equation with Robin and free boundary conditions, J. Differential Equations, 259 (2015), 423-453.  doi: 10.1016/j.jde.2015.02.012.  Google Scholar [25] D. Ludwig, D. G. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biol., 8 (1979), 217-258.  doi: 10.1007/BF00276310.  Google Scholar [26] L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in $\mathbb{R}^N$, J. Differential Equations, 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar [27] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar [28] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorem for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.  Google Scholar [29] N. Shigesada and K. Kawasaki, Biological Invasions; Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. Google Scholar [30] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar
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