April  2015, 35(4): 1531-1560. doi: 10.3934/dcds.2015.35.1531

Invasion entire solutions in a competition system with nonlocal dispersal

1. 

School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

3. 

School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received  July 2013 Revised  December 2013 Published  November 2014

This paper is concerned with invasion entire solutions of a Lotka-Volterra competition system with nonlocal dispersal, which formulate a new invasion way of the stronger species to the weaker one. We first give the asymptotic behavior of traveling wave solutions at infinity. Then by the comparison principle and sub-super solutions method, we establish the existence of invasion entire solutions which behave as two monotone waves with different speeds and coming from both sides of $x$-axis.
Citation: Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar

[2]

P. W. Bates, On some nonlocal evolution equations arising in materials science,, in H. Brunner, 48 (2006), 13. Google Scholar

[3]

P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[4]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[5]

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

[6]

X. Chen, J.-S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearity,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207. doi: 10.1017/S0308210500004959. Google Scholar

[7]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation,, Taiwanese J. Math., 8 (2004), 15. Google Scholar

[8]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153. Google Scholar

[9]

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. doi: 10.3934/dcds.2005.12.193. Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku. Math. J., 62 (2010), 17. doi: 10.2748/tmj/1270041024. Google Scholar

[11]

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. doi: 10.1016/j.jde.2012.01.009. Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2713. doi: 10.3934/dcdsb.2012.17.2713. Google Scholar

[13]

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

[14]

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

[15]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, Numerical and Applied Mathematics, (1989), 687. Google Scholar

[16]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233. doi: 10.1007/BF03167402. Google Scholar

[17]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. Google Scholar

[18]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar

[19]

C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[20]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[21]

W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[22]

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. doi: 10.1016/j.jde.2008.03.023. Google Scholar

[23]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,, J. Differential Equations, 244 (2008), 487. doi: 10.1016/j.jde.2007.10.019. Google Scholar

[24]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar

[25]

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

[26]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. doi: 10.1137/080723715. Google Scholar

[27]

J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,, Third edition. Interdisciplinary Applied Mathematics, 18 (2003). Google Scholar

[28]

S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y. Google Scholar

[29]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012), 1. doi: 10.1186/1687-2770-2012-120. Google Scholar

[30]

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. doi: 10.1016/j.jde.2011.04.020. Google Scholar

[31]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems,, Translation of Mathematical Monographs, (1994). Google Scholar

[32]

M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed,, Nonlinearity, 23 (2010), 1609. doi: 10.1088/0951-7715/23/7/005. Google Scholar

[33]

Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047. doi: 10.1090/S0002-9947-08-04694-1. Google Scholar

[34]

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. doi: 10.1137/080727312. Google Scholar

[35]

S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics,, Nonlinear Anal. Real World Appl., 13 (2012), 1991. doi: 10.1016/j.nonrwa.2011.12.020. Google Scholar

[36]

S.-L. Wu, Y.-J. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921. doi: 10.3934/dcds.2013.33.921. Google Scholar

[37]

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

[38]

Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications,, ANZIAM. J., 51 (2009), 49. doi: 10.1017/S1446181109000406. Google Scholar

[39]

Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems,, Chaos, 45 (2012), 1361. doi: 10.1016/j.chaos.2012.07.002. Google Scholar

[40]

G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021. Google Scholar

[41]

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. doi: 10.1016/j.jde.2012.01.014. Google Scholar

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar

[2]

P. W. Bates, On some nonlocal evolution equations arising in materials science,, in H. Brunner, 48 (2006), 13. Google Scholar

[3]

P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar

[4]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[5]

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

[6]

X. Chen, J.-S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearity,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207. doi: 10.1017/S0308210500004959. Google Scholar

[7]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation,, Taiwanese J. Math., 8 (2004), 15. Google Scholar

[8]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153. Google Scholar

[9]

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. doi: 10.3934/dcds.2005.12.193. Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku. Math. J., 62 (2010), 17. doi: 10.2748/tmj/1270041024. Google Scholar

[11]

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. doi: 10.1016/j.jde.2012.01.009. Google Scholar

[12]

J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2713. doi: 10.3934/dcdsb.2012.17.2713. Google Scholar

[13]

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

[14]

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

[15]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, Numerical and Applied Mathematics, (1989), 687. Google Scholar

[16]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233. doi: 10.1007/BF03167402. Google Scholar

[17]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. Google Scholar

[18]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar

[19]

C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[20]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[21]

W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[22]

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. doi: 10.1016/j.jde.2008.03.023. Google Scholar

[23]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,, J. Differential Equations, 244 (2008), 487. doi: 10.1016/j.jde.2007.10.019. Google Scholar

[24]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion equations,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar

[25]

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

[26]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. doi: 10.1137/080723715. Google Scholar

[27]

J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications,, Third edition. Interdisciplinary Applied Mathematics, 18 (2003). Google Scholar

[28]

S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y. Google Scholar

[29]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012), 1. doi: 10.1186/1687-2770-2012-120. Google Scholar

[30]

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. doi: 10.1016/j.jde.2011.04.020. Google Scholar

[31]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Sysytems,, Translation of Mathematical Monographs, (1994). Google Scholar

[32]

M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed,, Nonlinearity, 23 (2010), 1609. doi: 10.1088/0951-7715/23/7/005. Google Scholar

[33]

Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047. doi: 10.1090/S0002-9947-08-04694-1. Google Scholar

[34]

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. doi: 10.1137/080727312. Google Scholar

[35]

S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics,, Nonlinear Anal. Real World Appl., 13 (2012), 1991. doi: 10.1016/j.nonrwa.2011.12.020. Google Scholar

[36]

S.-L. Wu, Y.-J. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity,, Discrete Contin. Dyn. Syst., 33 (2013), 921. doi: 10.3934/dcds.2013.33.921. Google Scholar

[37]

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

[38]

Z. Yu and R. Yuan, Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications,, ANZIAM. J., 51 (2009), 49. doi: 10.1017/S1446181109000406. Google Scholar

[39]

Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems,, Chaos, 45 (2012), 1361. doi: 10.1016/j.chaos.2012.07.002. Google Scholar

[40]

G.-B. Zhang, W.-T. Li and Y.-J. Sun, Asymptotic behavior for nonlocal dispersal equations,, Nonlinear Anal., 72 (2010), 4466. doi: 10.1016/j.na.2010.02.021. Google Scholar

[41]

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. doi: 10.1016/j.jde.2012.01.014. Google Scholar

[1]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006

[2]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[3]

Fang-Di Dong, Wan-Tong Li, Jia-Bing Wang. Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272

[4]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160

[5]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[6]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423

[7]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053

[8]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178

[9]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107

[10]

Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075

[11]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011

[12]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[13]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[14]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969

[15]

Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084

[16]

Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099

[17]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707

[18]

Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779

[19]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023

[20]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]