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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2021, 41(9): 4013-4039.
doi: 10.3934/dcds.2021026
Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph
| 1. | Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan |
| 2. | Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan |
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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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983).
doi: 10.1090/memo/0285. |
| [8] |
J. W. Cahn,
Theory of crystal growth and interface motion in crystalline materials, Acta. Metall., 8 (1960), 554-562.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
K. P. Hadeler and F. Rothe,
Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
| [16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, Springer-Verlag, Berlin-New York, 1981. |
| [17] |
F. Hamel,
Bistable transition fronts in $\mathbb{R}^N$, Adv. Math., 289 (2016), 279-344.
doi: 10.1016/j.aim.2015.11.033. |
| [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. |
| [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. |
| [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. |
| [21] |
S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint. |
| [22] |
J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2$^nd$ edition, Springer, 2009.
doi: 10.1007/978-0-387-79388-7. |
| [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.
|
| [24] |
H. P. Jr. McKean,
Nagumo's equation, Adv. Math., 4 (1970), 209-223.
doi: 10.1016/0001-8708(70)90023-X. |
| [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. |
| [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. |
| [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. |
| [28] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. |
| [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. |
| [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. |
| [31] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983).
doi: 10.1090/memo/0285. |
| [8] |
J. W. Cahn,
Theory of crystal growth and interface motion in crystalline materials, Acta. Metall., 8 (1960), 554-562.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
K. P. Hadeler and F. Rothe,
Traveling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
| [16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, Springer-Verlag, Berlin-New York, 1981. |
| [17] |
F. Hamel,
Bistable transition fronts in $\mathbb{R}^N$, Adv. Math., 289 (2016), 279-344.
doi: 10.1016/j.aim.2015.11.033. |
| [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. |
| [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. |
| [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. |
| [21] |
S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint. |
| [22] |
J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2$^nd$ edition, Springer, 2009.
doi: 10.1007/978-0-387-79388-7. |
| [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.
|
| [24] |
H. P. Jr. McKean,
Nagumo's equation, Adv. Math., 4 (1970), 209-223.
doi: 10.1016/0001-8708(70)90023-X. |
| [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. |
| [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. |
| [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. |
| [28] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, 1997. |
| [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. |
| [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. |
| [31] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
| [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. |
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
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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