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September  2021, 41(9): 4013-4039. doi: 10.3934/dcds.2021026

Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Shuichi Jimbo 1, and  Yoshihisa Morita 2,,

1. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan

* Corresponding author: Yoshihisa Morita

Received  June 2020 Revised  December 2020 Published  September 2021 Early access  January 2021

Fund Project: The authors were partially supported by JSPS KAKENHI Grant Number JP18H01139 and the first author was supported in part by JSPS KAKENHI Grant Number JP16K05218

We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as $ t\to\pm\infty $ and classify the entire solutions according to their behaviors, where an entire solution is meant by a classical solution defined for all $ t\in(-\infty, \infty) $. To this end, we give a condition under that the front propagation is blocked by the emergence of standing stationary solutions. The existence of an entire solution which propagates beyond the blocking is also shown.

Keywords: Reaction-diffusion equation, bistable nonlinearity, entire solution, traveling wave, front propagation.
Mathematics Subject Classification: Primary: 35K57, 35B08; Secondary: 35B35, 35B40.
Citation: Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation, Partial Differential Equations and Related Topics (ed. J. A. Goldstein), Lecture Notes in Math, Springer, Berlin, Heidelberg, 1975.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. Berestyciki, J. Bouhours and G. Chapuisat, Front blocking and propagation in cylinders with varying cross section, Calc. Var., 55 (2016), Art. 44, 32 pp. doi: 10.1007/s00526-016-0962-2.  Google Scholar

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H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

[6]

H. Berestycki, F. Hamel and H. Matano, Bistable travelling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275.  Google Scholar

[7]

M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983). doi: 10.1090/memo/0285.  Google Scholar

[8]

J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta. Metall., 8 (1960), 554-562.   Google Scholar

[9]

X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[10]

Y. Du, B. Lou, R. Peng and M. Zhou, The Fisher-KPP equation over simple graphs: varied persistence states in river networks, J. Math. Biol., 80 (2020), 1559-1616.  doi: 10.1007/s00285-020-01474-1.  Google Scholar

[11]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[13]

H. Guo, F. Hamel and W.-J. Sheng, On the mean speed of bistable transition fronts in unbounded domains, J. Math. Pures Appl., 136 (2020), 92-157.  doi: 10.1016/j.matpur.2020.02.002.  Google Scholar

[14]

J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[15]

K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

F. Hamel, Bistable transition fronts in $\mathbb{R}^N$, Adv. Math., 289 (2016), 279-344.  doi: 10.1016/j.aim.2015.11.033.  Google Scholar

[18]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[19]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[20]

S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differential Equations, 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008.  Google Scholar

[21]

S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint. Google Scholar

[22]

J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2$^nd$ edition, Springer, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[23]

A. Kolmogorov, I. Petrovsky and and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sec. A, 1 (1937), 1-26.   Google Scholar

[24]

H. P. Jr. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223.  doi: 10.1016/0001-8708(70)90023-X.  Google Scholar

[25]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[26]

J. Nagumo, S. Yoshizawa and S. Arimoto, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[27]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[28]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. Google Scholar

[29]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[30]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[31]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[32]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation, Partial Differential Equations and Related Topics (ed. J. A. Goldstein), Lecture Notes in Math, Springer, Berlin, Heidelberg, 1975.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

H. Berestyciki, J. Bouhours and G. Chapuisat, Front blocking and propagation in cylinders with varying cross section, Calc. Var., 55 (2016), Art. 44, 32 pp. doi: 10.1007/s00526-016-0962-2.  Google Scholar

[5]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.  Google Scholar

[6]

H. Berestycki, F. Hamel and H. Matano, Bistable travelling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275.  Google Scholar

[7]

M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983). doi: 10.1090/memo/0285.  Google Scholar

[8]

J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta. Metall., 8 (1960), 554-562.   Google Scholar

[9]

X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[10]

Y. Du, B. Lou, R. Peng and M. Zhou, The Fisher-KPP equation over simple graphs: varied persistence states in river networks, J. Math. Biol., 80 (2020), 1559-1616.  doi: 10.1007/s00285-020-01474-1.  Google Scholar

[11]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[13]

H. Guo, F. Hamel and W.-J. Sheng, On the mean speed of bistable transition fronts in unbounded domains, J. Math. Pures Appl., 136 (2020), 92-157.  doi: 10.1016/j.matpur.2020.02.002.  Google Scholar

[14]

J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[15]

K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

F. Hamel, Bistable transition fronts in $\mathbb{R}^N$, Adv. Math., 289 (2016), 279-344.  doi: 10.1016/j.aim.2015.11.033.  Google Scholar

[18]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[19]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[20]

S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differential Equations, 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008.  Google Scholar

[21]

S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint. Google Scholar

[22]

J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2$^nd$ edition, Springer, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[23]

A. Kolmogorov, I. Petrovsky and and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bjul. Moskowskogo Gos. Univ. Ser. Internat. Sec. A, 1 (1937), 1-26.   Google Scholar

[24]

H. P. Jr. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223.  doi: 10.1016/0001-8708(70)90023-X.  Google Scholar

[25]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[26]

J. Nagumo, S. Yoshizawa and S. Arimoto, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Eng., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[27]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[28]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. Google Scholar

[29]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[30]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar

[31]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[32]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

Figure 1.  An example of the domain $ \Omega $ consisting of $ \Omega_k\; (1\leq k\leq5) $. $ \Omega_i\; (i = 1,2) $ are wave incoming branches, where the arrows in the figure indicate the directions of the front propagation
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Figure 2.  Simplified dynamics around the stationary solutions for the cases $ (a)\; F(1)+(\kappa^2-1)F(a)<0,\; (b)\; F(1)+(\kappa^2-1)F(a) = 0, $ and $ (c)\; F(1)+(\kappa^2-1)F(a)>0 $. Although the phase space is infinite-dimensional, the figures indicate one-dimensional dynamics on the dominant subspace. $ \hat{u}, \hat{u}^{(1)}, \hat{u}^{(2)} $ and $ \hat{u}^{(12)} $ stand for the entire solutions stated in Theorem 1.1
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Figure 3.  Case: $ m = 4,\; d_1 = d_2 = d_3 = d_4. $ (a) $ \kappa = 9 $, and $ F(1)+(\kappa^2-1)F(a)<0 $. An incoming front in $ \Omega_1 $ is blocked. (b) $ \kappa^+ = 0 $, so $ F(1)+((\kappa^+)^2-1)F(a)>0 $. Two incoming fronts from $ \Omega_1 $ and $ \Omega_2 $ go through the junction and propagate to the branches $ \Omega_3 $ and $ \Omega_4 $
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