# American Institute of Mathematical Sciences

May  2019, 18(3): 1049-1072. doi: 10.3934/cpaa.2019051

## Entire solutions in nonlocal monostable equations: Asymmetric case

 1 State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, Shaanxi 710071, China 2 School of Science, Chang'an University, Xi'an, Shaanxi 710064, China 3 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  November 2017 Revised  August 2018 Published  November 2018

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., $u_{t}=J*u-u+f(u)$. Here the kernel $J$ is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds $c^{*}$ and $\hat{c}^{*}$, where $c^*$ and $\hat{c}^{*}$ are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel $J$ is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

Citation: Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051
##### References:
 [1] F. Andreuvaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. Toledomelero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [2] P. W. Bates, On some nonlocal evolution equations arising in materials science, in H. Brunner, X.Q. Zhao and X. Zou (Eds.), Nonlinear dynamics and evolution equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13–52. doi: 10.1090/fic/048/02. [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), 105-136.  doi: 10.1007/s002050050037. [4] 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. [5] E. Chasseigne, M. Chavesb and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005. [6] X. F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal. TMA, 50 (2002), 807-838.  doi: 10.1016/S0362-546X(01)00787-8. [7] F. X. Chen, Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [8] F. 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), 1207-1237.  doi: 10.1017/S0308210500004959. [9] C. Cortazar, M. Elgueta, J. D Rossi and N. Wolanski, Boundary fluxes for non-local diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [10] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [11] J. Coville, Traveling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Prépublication du CMM, Hal-00696208. [12] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002. [13] J. Coville and L. Dupaigne, On a nonlocal reaction-diffusion eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. [14] J. Coville, Travelling waves in a nonlocal reaction diffusion equation with ignition nonlinearity, Ph.D. Thesis, Paris: Universit'e Pierre et Marie Curie, 2003. [15] J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differential Equations, 68 (2007), 1-23. [16] F. D. Dong, W. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272. [17] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. doi: 10.1007/978-3-662-05281-5_3. [18] F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution, 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. [19] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [20] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 211-232.  doi: 10.1017/S0956792506006462. [21] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [22] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Functional Analysis, 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [23] T. Kawata, Fourier Analysi, Sangyo Tosho Publishing Co. LTD, Tokyo, 1975. [24] T. S. Lim and A. Alatos, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602. [25] W. T. Li, N. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.  doi: 10.1016/j.matpur.2008.07.002. [26] 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), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005. [27] 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. [28] 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. [29] Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x. [30] 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. [31] S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y. [32] K. Schumacher, Traveling-front solutions for integro-differential equations, I, J. Reine. Angew. Math., 316 (1980), 54-70. [33] 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. [34] Y. J. Sun, W. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonnlinearity, Nonlinear Anal. TMA., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032. [35] A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003. [36] M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005. [37] 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. [38] 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. [39] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028. [40] S. L. Wu, Z. X. Shi and F. Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049. [41] 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. [42] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648. [43] H. Yagisita, Existence of traveling waves for a nonlocal bistable equation: an abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.  doi: 10.2977/prims/1260476649. [44] L. Zhang, W. T. Li and Z. C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804. doi: 10.1007/s11425-016-9003-7. [45] 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] F. Andreuvaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. Toledomelero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [2] P. W. Bates, On some nonlocal evolution equations arising in materials science, in H. Brunner, X.Q. Zhao and X. Zou (Eds.), Nonlinear dynamics and evolution equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13–52. doi: 10.1090/fic/048/02. [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), 105-136.  doi: 10.1007/s002050050037. [4] 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. [5] E. Chasseigne, M. Chavesb and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005. [6] X. F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal. TMA, 50 (2002), 807-838.  doi: 10.1016/S0362-546X(01)00787-8. [7] F. X. Chen, Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [8] F. 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), 1207-1237.  doi: 10.1017/S0308210500004959. [9] C. Cortazar, M. Elgueta, J. D Rossi and N. Wolanski, Boundary fluxes for non-local diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [10] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [11] J. Coville, Traveling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Prépublication du CMM, Hal-00696208. [12] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002. [13] J. Coville and L. Dupaigne, On a nonlocal reaction-diffusion eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. [14] J. Coville, Travelling waves in a nonlocal reaction diffusion equation with ignition nonlinearity, Ph.D. Thesis, Paris: Universit'e Pierre et Marie Curie, 2003. [15] J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differential Equations, 68 (2007), 1-23. [16] F. D. Dong, W. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272. [17] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. doi: 10.1007/978-3-662-05281-5_3. [18] F. Hamel and N. Nadirashvili, Entire solution of the KPP eqution, 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. [19] F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Rational Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238. [20] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 211-232.  doi: 10.1017/S0956792506006462. [21] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [22] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Functional Analysis, 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [23] T. Kawata, Fourier Analysi, Sangyo Tosho Publishing Co. LTD, Tokyo, 1975. [24] T. S. Lim and A. Alatos, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615-8631.  doi: 10.1090/tran/6602. [25] W. T. Li, N. W. Liu and Z. C. Wang, Entire solutions in reaction-advection-diffusion equations in cylinders, J. Math. Pures Appl., 90 (2008), 492-504.  doi: 10.1016/j.matpur.2008.07.002. [26] 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), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005. [27] 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. [28] 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. [29] Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x. [30] 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. [31] S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y. [32] K. Schumacher, Traveling-front solutions for integro-differential equations, I, J. Reine. Angew. Math., 316 (1980), 54-70. [33] 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. [34] Y. J. Sun, W. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonnlinearity, Nonlinear Anal. TMA., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032. [35] A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003. [36] M. Wang and G. Lv, Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayed, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005. [37] 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. [38] 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. [39] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028. [40] S. L. Wu, Z. X. Shi and F. Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049. [41] 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. [42] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648. [43] H. Yagisita, Existence of traveling waves for a nonlocal bistable equation: an abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.  doi: 10.2977/prims/1260476649. [44] L. Zhang, W. T. Li and Z. C. Wang, Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel, Sci. China Math., 60 (2017), 1791-1804. doi: 10.1007/s11425-016-9003-7. [45] 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.
Region of $(c, \hat{c})$
 $\hat{c}^{*}>0$ $\hat{c}^{*}=0$ $\hat{c}^{*} <0$ ${c^{\ast}>0}$ $C_{11}$ $C_{12}$ $C_{13}=C^{1}_{13}\cup C^{2}_{13}$ ${c^{\ast}=0}$ $C_{21}$ $C_{22}$ $C_{23}=C^{1}_{23}\cup C^{2}_{23}$ ${c^{\ast} <0}$ $C_{31}=C^{1}_{31}\cup C^{2}_{31}$ $C_{32}=C^{1}_{32}\cup C^{2}_{32}$ $\setminus$
 $\hat{c}^{*}>0$ $\hat{c}^{*}=0$ $\hat{c}^{*} <0$ ${c^{\ast}>0}$ $C_{11}$ $C_{12}$ $C_{13}=C^{1}_{13}\cup C^{2}_{13}$ ${c^{\ast}=0}$ $C_{21}$ $C_{22}$ $C_{23}=C^{1}_{23}\cup C^{2}_{23}$ ${c^{\ast} <0}$ $C_{31}=C^{1}_{31}\cup C^{2}_{31}$ $C_{32}=C^{1}_{32}\cup C^{2}_{32}$ $\setminus$
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