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Divergence points in systems satisfying the specification property
Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity
1. | Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071 |
2. | Department of Mathematics, Xidian University, Xi'an, Shaanxi 710071, China |
3. | Department of Applied Mathematics, Xidian University, Xi'an 710071 |
References:
[1] |
V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184.
doi: 10.1007/BF00275212. |
[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 nonlinearity, Proc. R. Soc. Edinb. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[6] |
J. Fang and X. Q. Zhao, Monotone wavefronts for partially degenerate reaction diffusion systems, J. Dynam. Diff. Eqns., 21 (2009), 663-680.
doi: 10.1007/s10884-009-9152-7. |
[7] |
Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32. |
[8] |
Y. J. L. Guo, Entire solutions for a discrete diffusive equation, J. Math. Anal. Appl., 347 (2008), 450-458.
doi: 10.1016/j.jmaa.2008.03.076. |
[9] |
J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an applicationto discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212. |
[10] |
J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[11] |
K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Canad. Appl. Math. Quart., 10 (2002), 473-499. |
[12] |
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>3.3.CO;2-N. |
[13] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
M. A. Lewis and G. Schmitz, Biological invasion of an organism withseparate mobile and stationary states: modeling and analysis, Forma, 11 (1996), 1-25. |
[15] |
B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion system, J. Differential Equations, 252 (2012), 4842-4861.
doi: 10.1016/j.jde.2012.01.018. |
[16] |
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. |
[17] |
W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-III functional response, Chaos, Solitons and Fractals, 7 (2008), 476-486. |
[18] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.1090/S0002-9947-1990-0967316-X. |
[19] |
Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Diff. Eqns., 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[20] |
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. |
[21] |
S. Ruan and J. Wu, Reaction-diffusion equations with inifite delays, Canad. Appl. Math. Quart., 2 (1994), 485-550. |
[22] |
M. X. Wang and G. Y. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[23] |
Z. C. Wang, W. T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 563-607.
doi: 10.1007/s10884-008-9103-8. |
[24] |
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), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[25] |
Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[26] |
Z. C. Wang and W. T. Li, Dynamics of a nonlocal delayed reaction-diffusion equation without quasi-monotonicity, Proc. R. Soc. Edinb. A, 140 (2010), 1081-1109.
doi: 10.1017/S0308210509000262. |
[27] |
S. L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA, 13 (2012), 1991-2005. |
[28] |
D. Xu and X. Q. Zhao, Erratum to "Bistable waves in an epidemic model'', J. Dynam. Diff. Eqns., 17 (2005), 219-247.
doi: 10.1007/s10884-005-6294-0. |
[29] |
H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[30] |
K. Zhang and X. Q. Zhao, Asymptotic behavior of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. Lond. A, 463 (2007), 1029-1043.
doi: 10.1098/rspa.2006.1806. |
[31] |
P. A. Zhang and W. T. Li, Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal. TMA, 72 (2010), 2178-2189.
doi: 10.1016/j.na.2009.10.016. |
[32] |
X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128. |
show all references
References:
[1] |
V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184.
doi: 10.1007/BF00275212. |
[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 nonlinearity, Proc. R. Soc. Edinb. A, 136 (2006), 1207-1237.
doi: 10.1017/S0308210500004959. |
[6] |
J. Fang and X. Q. Zhao, Monotone wavefronts for partially degenerate reaction diffusion systems, J. Dynam. Diff. Eqns., 21 (2009), 663-680.
doi: 10.1007/s10884-009-9152-7. |
[7] |
Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32. |
[8] |
Y. J. L. Guo, Entire solutions for a discrete diffusive equation, J. Math. Anal. Appl., 347 (2008), 450-458.
doi: 10.1016/j.jmaa.2008.03.076. |
[9] |
J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an applicationto discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212. |
[10] |
J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.
doi: 10.2748/tmj/1270041024. |
[11] |
K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Canad. Appl. Math. Quart., 10 (2002), 473-499. |
[12] |
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>3.3.CO;2-N. |
[13] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation, Arch. Ration. Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[14] |
M. A. Lewis and G. Schmitz, Biological invasion of an organism withseparate mobile and stationary states: modeling and analysis, Forma, 11 (1996), 1-25. |
[15] |
B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion system, J. Differential Equations, 252 (2012), 4842-4861.
doi: 10.1016/j.jde.2012.01.018. |
[16] |
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. |
[17] |
W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-III functional response, Chaos, Solitons and Fractals, 7 (2008), 476-486. |
[18] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.1090/S0002-9947-1990-0967316-X. |
[19] |
Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Diff. Eqns., 18 (2006), 841-861.
doi: 10.1007/s10884-006-9046-x. |
[20] |
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. |
[21] |
S. Ruan and J. Wu, Reaction-diffusion equations with inifite delays, Canad. Appl. Math. Quart., 2 (1994), 485-550. |
[22] |
M. X. Wang and G. Y. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[23] |
Z. C. Wang, W. T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 563-607.
doi: 10.1007/s10884-008-9103-8. |
[24] |
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), 2047-2084.
doi: 10.1090/S0002-9947-08-04694-1. |
[25] |
Z. C. Wang, W. T. Li and S. Ruan, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.
doi: 10.1137/080727312. |
[26] |
Z. C. Wang and W. T. Li, Dynamics of a nonlocal delayed reaction-diffusion equation without quasi-monotonicity, Proc. R. Soc. Edinb. A, 140 (2010), 1081-1109.
doi: 10.1017/S0308210509000262. |
[27] |
S. L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA, 13 (2012), 1991-2005. |
[28] |
D. Xu and X. Q. Zhao, Erratum to "Bistable waves in an epidemic model'', J. Dynam. Diff. Eqns., 17 (2005), 219-247.
doi: 10.1007/s10884-005-6294-0. |
[29] |
H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.
doi: 10.2977/prims/1145476150. |
[30] |
K. Zhang and X. Q. Zhao, Asymptotic behavior of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. Lond. A, 463 (2007), 1029-1043.
doi: 10.1098/rspa.2006.1806. |
[31] |
P. A. Zhang and W. T. Li, Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal. TMA, 72 (2010), 2178-2189.
doi: 10.1016/j.na.2009.10.016. |
[32] |
X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128. |
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