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February  2013, 33(2): 921-946. doi: 10.3934/dcds.2013.33.921

Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity

Shi-Liang Wu 1, , Yu-Juan Sun 2, and  San-Yang Liu 3,

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

Received  August 2011 Revised  July 2012 Published  September 2012

This paper is concerned with traveling fronts and entire solutions for a class of monostable partially degenerate reaction-diffusion systems. It is known that the system admits traveling wave solutions. In this paper, we first prove the monotonicity and uniqueness of the traveling wave solutions, and the existence of spatially independent solutions. Combining traveling fronts with different speeds and a spatially independent solution, the existence and various qualitative features of entire solutions are then established by using comparison principle. As applications, we consider a reaction-diffusion model with a quiescent stage in population dynamics and a man-environment-man epidemic model in physiology.
Keywords: traveling front, monostable nonlinearity., Entire solution, spatially independent solution, partially degenerate reaction-diffusion system.
Mathematics Subject Classification: Primary: 35K57, 35B05, 35B40; Secondary: 92D25, 92D3.
Citation: Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[7]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.  Google Scholar

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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.  Google Scholar

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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.  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), 17-28. doi: 10.2748/tmj/1270041024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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M. A. Lewis and G. Schmitz, Biological invasion of an organism withseparate mobile and stationary states: modeling and analysis, Forma, 11 (1996), 1-25.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Ruan and J. Wu, Reaction-diffusion equations with inifite delays, Canad. Appl. Math. Quart., 2 (1994), 485-550.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA, 13 (2012), 1991-2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.  Google Scholar

[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.  Google Scholar

[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.  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), 17-28. doi: 10.2748/tmj/1270041024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Ruan and J. Wu, Reaction-diffusion equations with inifite delays, Canad. Appl. Math. Quart., 2 (1994), 485-550.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA, 13 (2012), 1991-2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.  Google Scholar

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