Advanced Search
Article Contents
Article Contents

Entire solutions in nonlocal monostable equations: Asymmetric case

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35K57, 35B08; Secondary: 34C14.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  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}$ $\setminus$
     | Show Table
    DownLoad: CSV
  •   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.
      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.
      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.
      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.
      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.
      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.
      F. X. Chen , Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations, Adv. Differential Equations, 2 (1997) , 125-160. 
      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.
      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.
      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.
      J. Coville, Traveling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Prépublication du CMM, Hal-00696208.
      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.
      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. 
      J. Coville, Travelling waves in a nonlocal reaction diffusion equation with ignition nonlinearity, Ph.D. Thesis, Paris: Universit'e Pierre et Marie Curie, 2003.
      J. Coville , Maximum principles, sliding techniques and applications to nonlocal equations, Electron. J. Differential Equations, 68 (2007) , 1-23. 
      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.
      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.
      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.
      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.
      V. Hutson  and  M. Grinfeld , Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006) , 211-232.  doi: 10.1017/S0956792506006462.
      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.
      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.
      T. Kawata, Fourier Analysi, Sangyo Tosho Publishing Co. LTD, Tokyo, 1975.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      K. Schumacher , Traveling-front solutions for integro-differential equations, I, J. Reine. Angew. Math., 316 (1980) , 54-70. 
      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.
      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.
      A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003.
      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.
      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.
      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.
      H. F. Weinberger , Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982) , 353-396.  doi: 10.1137/0513028.
      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.
      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.
      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.
      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.
      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.
      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.
  • 加载中



Article Metrics

HTML views(1893) PDF downloads(415) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint