# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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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##### 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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