# 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 & 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. Google Scholar [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. 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), 105-136. 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), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [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. Google Scholar [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. Google Scholar [7] F. X. Chen, Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] J. Coville, Traveling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Prépublication du CMM, Hal-00696208.Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [15] J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differential Equations, 68 (2007), 1-23. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 211-232. doi: 10.1017/S0956792506006462. Google Scholar [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. Google Scholar [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. Google Scholar [23] T. Kawata, Fourier Analysi, Sangyo Tosho Publishing Co. LTD, Tokyo, 1975.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] K. Schumacher, Traveling-front solutions for integro-differential equations, I, J. Reine. Angew. Math., 316 (1980), 54-70. Google Scholar [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. Google Scholar [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. Google Scholar [35] A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. 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), 105-136. 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), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar [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. Google Scholar [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. Google Scholar [7] F. X. Chen, Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] J. Coville, Traveling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Prépublication du CMM, Hal-00696208.Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [15] J. Coville, Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differential Equations, 68 (2007), 1-23. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 211-232. doi: 10.1017/S0956792506006462. Google Scholar [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. Google Scholar [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. Google Scholar [23] T. Kawata, Fourier Analysi, Sangyo Tosho Publishing Co. LTD, Tokyo, 1975.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] K. Schumacher, Traveling-front solutions for integro-differential equations, I, J. Reine. Angew. Math., 316 (1980), 54-70. Google Scholar [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. Google Scholar [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. Google Scholar [35] A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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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