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June  2020, 40(6): 3375-3394. doi: 10.3934/dcds.2020033

Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity

Maho Endo , Yuki Kaneko 1**, and  Yoshio Yamada 2,,

Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

* Corresponding author: Yoshio Yamada

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday
1 Partially supported by Waseda University Grant for Special Research Project (2018K-203, 2018B-103) and Grant-in-Aid for Early-Career Scientists (19K14602).
2 Partially supported by Grant-in-Aid for Scientific Research (C) 16K05244.
** Current address: Department of Mathematical and Physical Sciences, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681, Japan.

Received  January 2019 Revised  August 2019 Published  October 2019

This paper deals with a free boundary problem for a reaction-diffusion equation in a one-dimensional interval whose boundary consists of a fixed end-point and a moving one. We put homogeneous Dirichlet condition at the fixed boundary, while we assume that the dynamics of the moving boundary is governed by the Stefan condition. Such free boundary problems have been studied by a lot of researchers. We will take a nonlinear reaction term of positive bistable type which exhibits interesting properties of solutions such as multiple spreading phenomena. In fact, it will be proved that large-time behaviors of solutions can be classified into three types; vanishing, small spreading and big spreading. Some sufficient conditions for these behaviors are also shown. Moreover, for two types of spreading, we will give sharp estimates of spreading speed of each free boundary and asymptotic profiles of each solution.

Keywords: Free boundary problem, spreading speed, asymptotic profile, reaction-diffusion equation.
Mathematics Subject Classification: Primary: 35R35; Secondary: 35K57, 35J61, 92D25.
Citation: Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.

[2]

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.

[3]

W. Choi and I. Ahn, Non-uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment, J. Math. Anal. Appl., 479 (2019), 283-314.  doi: 10.1016/j.jmaa.2019.06.027.

[4]

W. Ding, R. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.

[5]

Y. Du and Z. G. 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: 10.1137/090771089.

[6]

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.

[7]

Y. Du, B. Lou and M. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed, and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848.

[8]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312.  doi: 10.4171/JEMS/198.

[9]

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.

[10]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[11]

Y. Kaneko, K. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations, Funkcial. Ekvac., 57 (2014), 449-465.  doi: 10.1619/fesi.57.449.

[12]

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. 

[13]

Y. Kaneko and Y. Yamada, Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions, J. Math. Anal. Appl., 465 (2018), 1159-1175.  doi: 10.1016/j.jmaa.2018.05.056.

[14]

Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differential Equations, 265 (2018), 1000-1043.  doi: 10.1016/j.jde.2018.03.026.

[15]

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, to appear in SIAM J. Math. Anal.

[16]

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.

[17]

C. Lei, H. Matsuzawa, R. Peng and M. Zhou, Refined estimates for the propagation speed of the transition solution to a free boundary problem with a nonlinearity of combustion type, J. Differential Equations, 265 (2018), 2897-2920.  doi: 10.1016/j.jde.2018.04.053.

[18]

X. Liu and B. Lou, Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions, Math. Model. Nat. Phenom., 8 (2013), 18-32.  doi: 10.1051/mmnp/20138303.

[19]

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.

[20]

B. Lou, Convergence in time-periodic quasilinear parabolic equations in one space dimension, J. Differential Equations, 265 (2018), 3952-3969.  doi: 10.1016/j.jde.2018.05.025.

[21]

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.

[22]

H. Matsuzawa, A free boundary problem for the Fisher-KPP equation with a given moving boundary, Commun. Pure Appl. Anal., 17 (2018), 1821-1852.  doi: 10.3934/cpaa.2018087.

[23]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  doi: 10.1512/iumj.1972.21.21079.

[24]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.  doi: 10.1016/0022-0396(81)90077-2.

[25]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Art. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.

[26]

R. H. Wang, L. Wang and Z. C. Wang, Free boundary problem of a reaction-diffusion equation with nonlinear convection term, J. Math. Anal. Appl., 467 (2018), 1233-1257.  doi: 10.1016/j.jmaa.2018.07.065.

[27]

Y. Zhao and M. X. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.

show all references

References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.

[2]

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.

[3]

W. Choi and I. Ahn, Non-uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment, J. Math. Anal. Appl., 479 (2019), 283-314.  doi: 10.1016/j.jmaa.2019.06.027.

[4]

W. Ding, R. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779.  doi: 10.1016/j.jde.2017.04.013.

[5]

Y. Du and Z. G. 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: 10.1137/090771089.

[6]

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.

[7]

Y. Du, B. Lou and M. Zhou, Nonlinear diffusion problems with free boundaries: Convergence, transition speed, and zero number arguments, SIAM J. Math. Anal., 47 (2015), 3555-3584.  doi: 10.1137/140994848.

[8]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312.  doi: 10.4171/JEMS/198.

[9]

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.

[10]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[11]

Y. Kaneko, K. Oeda and Y. Yamada, Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations, Funkcial. Ekvac., 57 (2014), 449-465.  doi: 10.1619/fesi.57.449.

[12]

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. 

[13]

Y. Kaneko and Y. Yamada, Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions, J. Math. Anal. Appl., 465 (2018), 1159-1175.  doi: 10.1016/j.jmaa.2018.05.056.

[14]

Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differential Equations, 265 (2018), 1000-1043.  doi: 10.1016/j.jde.2018.03.026.

[15]

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, to appear in SIAM J. Math. Anal.

[16]

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.

[17]

C. Lei, H. Matsuzawa, R. Peng and M. Zhou, Refined estimates for the propagation speed of the transition solution to a free boundary problem with a nonlinearity of combustion type, J. Differential Equations, 265 (2018), 2897-2920.  doi: 10.1016/j.jde.2018.04.053.

[18]

X. Liu and B. Lou, Asymptotic behavior of solutions to diffusion problems with Robin and free boundary conditions, Math. Model. Nat. Phenom., 8 (2013), 18-32.  doi: 10.1051/mmnp/20138303.

[19]

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.

[20]

B. Lou, Convergence in time-periodic quasilinear parabolic equations in one space dimension, J. Differential Equations, 265 (2018), 3952-3969.  doi: 10.1016/j.jde.2018.05.025.

[21]

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.

[22]

H. Matsuzawa, A free boundary problem for the Fisher-KPP equation with a given moving boundary, Commun. Pure Appl. Anal., 17 (2018), 1821-1852.  doi: 10.3934/cpaa.2018087.

[23]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.  doi: 10.1512/iumj.1972.21.21079.

[24]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.  doi: 10.1016/0022-0396(81)90077-2.

[25]

N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advections, Calc. Var. Partial Differential Equations, 56 (2017), Art. 61, 36 pp. doi: 10.1007/s00526-017-1165-1.

[26]

R. H. Wang, L. Wang and Z. C. Wang, Free boundary problem of a reaction-diffusion equation with nonlinear convection term, J. Math. Anal. Appl., 467 (2018), 1233-1257.  doi: 10.1016/j.jmaa.2018.07.065.

[27]

Y. Zhao and M. X. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.

Figure 1.  The phase plane of $ \rm(SP) $
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