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Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions
Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of $x$-axis for a three-species Lotka-Volterra competition system.
References:
[1] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
[2] |
J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[3] |
X. Chen and J. S. Guo,
Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.
doi: 10.1007/s00208-003-0414-0. |
[4] |
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. |
[5] |
X. Chen, J. S. Guo and H. Ninomiya,
Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[6] |
X. Chen, S. C. Fu and J. S. Guo,
Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
doi: 10.1137/050627824. |
[7] |
A. De Masi, T. Gobron and E. Presutti,
Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.
doi: 10.1007/BF00380506. |
[8] |
A. De Masi, E. Orlandi, E. Presutti and L. Triolo,
Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.
doi: 10.1017/S0308210500022472. |
[9] |
O. Diekmann and H. Kaper,
On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
doi: 10.1016/0362-546X(78)90015-9. |
[10] |
B. Ermentrout and J. Mcleod,
Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.
doi: 10.1017/S030821050002583X. |
[11] |
P. C. Fife,
Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191.
|
[12] |
P. C. Fife,
Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979. |
[13] |
Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.
doi: 10.11650/twjm/1500558454. |
[14] |
J. S. Guo and C. H. Wu,
Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.
doi: 10.1016/j.jde.2012.01.009. |
[15] |
J. S. Guo, Y. Wang, C. H. Wu and C. C. Wu,
The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.
doi: 10.11650/tjm.19.2015.5373. |
[16] |
J. S. Guo and Y. C. Lin,
Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629.
|
[17] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[18] |
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. |
[19] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equations, 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. |
[20] |
F. Hamel and N. Nadirashvili,
Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[21] |
W. T. Li, Y. J. Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[22] |
W. T. Li, J. B. Wang and L. Zhang,
Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.
doi: 10.1016/j.jde.2016.05.006. |
[23] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[24] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[25] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[26] |
Y. Morita and H. Ninomiya,
Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[27] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[28] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.
doi: 10.1016/j.jde.2011.04.020. |
[29] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[30] |
Z. C. Wang, W. T. Li and S. Ruan,
Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.
doi: 10.1007/s10884-008-9103-8. |
[31] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[32] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[33] |
C. H. Wu,
A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.
doi: 10.1007/s10884-016-9524-8. |
[34] |
S. L. Wu and H. Wang,
Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.
doi: 10.1007/s10884-013-9293-6. |
[35] |
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. |
[36] |
G. B. Zhang, W. T. Li and Z. C. Wang,
Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.
doi: 10.1016/j.jde.2012.01.014. |
[37] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
show all references
References:
[1] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
[2] |
J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[3] |
X. Chen and J. S. Guo,
Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.
doi: 10.1007/s00208-003-0414-0. |
[4] |
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. |
[5] |
X. Chen, J. S. Guo and H. Ninomiya,
Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. R. Soc. Edinb. Sect. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[6] |
X. Chen, S. C. Fu and J. S. Guo,
Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
doi: 10.1137/050627824. |
[7] |
A. De Masi, T. Gobron and E. Presutti,
Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132 (1995), 143-205.
doi: 10.1007/BF00380506. |
[8] |
A. De Masi, E. Orlandi, E. Presutti and L. Triolo,
Stability of the interface in a model of phase separation, Proc. R.Soc. Edinb. A, 124 (1994), 1013-1022.
doi: 10.1017/S0308210500022472. |
[9] |
O. Diekmann and H. Kaper,
On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
doi: 10.1016/0362-546X(78)90015-9. |
[10] |
B. Ermentrout and J. Mcleod,
Existence and uniqueness of travelling waves for a neural network, Proc. R. Soc. Edinb. A, 123 (1993), 461-478.
doi: 10.1017/S030821050002583X. |
[11] |
P. C. Fife,
Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153-191.
|
[12] |
P. C. Fife,
Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer, Berlin, 1979. |
[13] |
Y. Fukao, Y. Morita and H. Ninomiya,
Some entire solutions of the Allen-Cahn equation, Taiwanes J. Math., 8 (2004), 15-32.
doi: 10.11650/twjm/1500558454. |
[14] |
J. S. Guo and C. H. Wu,
Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.
doi: 10.1016/j.jde.2012.01.009. |
[15] |
J. S. Guo, Y. Wang, C. H. Wu and C. C. Wu,
The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.
doi: 10.11650/tjm.19.2015.5373. |
[16] |
J. S. Guo and Y. C. Lin,
Entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity, Osaka J. Math., 50 (2013), 607-629.
|
[17] |
J. S. Guo and C. H. Wu,
Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[18] |
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. |
[19] |
F. Hamel and N. Nadirashvili,
Entire solutions of the KPP equations, 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. |
[20] |
F. Hamel and N. Nadirashvili,
Traveling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[21] |
W. T. Li, Y. J. Sun and Z. C. Wang,
Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.
doi: 10.1016/j.nonrwa.2009.07.005. |
[22] |
W. T. Li, J. B. Wang and L. Zhang,
Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.
doi: 10.1016/j.jde.2016.05.006. |
[23] |
W. T. Li, Z. C. Wang and J. Wu,
Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.
doi: 10.1016/j.jde.2008.03.023. |
[24] |
W. T. Li, L. Zhang and G. B. Zhang,
Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.
doi: 10.3934/dcds.2015.35.1531. |
[25] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion equations, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[26] |
Y. Morita and H. Ninomiya,
Entire solution with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[27] |
Y. Morita and K. Tachibana,
An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[28] |
Y. J. Sun, W. T. Li and Z. C. Wang,
Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.
doi: 10.1016/j.jde.2011.04.020. |
[29] |
M. Wang and G. Lv,
Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlineariry, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[30] |
Z. C. Wang, W. T. Li and S. Ruan,
Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.
doi: 10.1007/s10884-008-9103-8. |
[31] |
Z. C. Wang, W. T. Li and S. Ruan,
Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Am. Math. Soc., 361 (2009), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[32] |
Z. C. Wang, W. T. Li and J. Wu,
Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[33] |
C. H. Wu,
A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.
doi: 10.1007/s10884-016-9524-8. |
[34] |
S. L. Wu and H. Wang,
Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.
doi: 10.1007/s10884-013-9293-6. |
[35] |
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. |
[36] |
G. B. Zhang, W. T. Li and Z. C. Wang,
Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.
doi: 10.1016/j.jde.2012.01.014. |
[37] |
L. Zhang, W. T. Li and S. L. Wu,
Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.
doi: 10.1007/s10884-014-9416-8. |
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