Advanced Search
Article Contents
Article Contents

Finite mass solutions for a nonlocal inhomogeneous dispersal equation

Abstract Related Papers Cited by
  • In this paper we study the asymptotic behavior of the following nonlocal inhomogeneous dispersal equation $$ u_t(x,t) = \int_{\mathbb{R}} J\left(\frac{x-y}{g(y)}\right) \frac{u(y,t)}{g(y)} dy -u(x,t) \qquad x\in \mathbb{R},\ t>0, $$ where $J$ is an even, smooth, probability density, and $g$, which accounts for a dispersal distance, is continuous and positive. We prove that if $g(|y|)\sim a |y|$ as $|y|\to + \infty$ for some $0 < a < 1$, there exists a unique (up to normalization) positive stationary solution, which is in $L^1(\mathbb{R})$. On the other hand, if $g(|y|)\sim |y|^p$, with $p > 2$ there are no positive stationary solutions. We also establish the asymptotic behavior of the solutions of the evolution problem in both cases.
    Mathematics Subject Classification: 45A05, 45J05.


    \begin{equation} \\ \end{equation}
  • [1]

    E. Chasseigne, M. Chaves 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.


    C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A non local inhomogeneous dispersal process, J. Diff. Eqns., 241 (2007), 332-358.doi: 10.1016/j.jde.2007.06.002.


    C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems, SIAM J. Math. Anal., 41 (2009), 2136-2164.doi: 10.1137/090751682.


    C. Cortázar, M. Elgueta, S. Martínez and J. Rossi, Random walks and the porous medium equation, Rev. Un. Mat. Argentina, 50 (2009), 149-155.


    J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953.doi: 10.1016/j.jde.2010.07.003.


    J. Coville, Harnack type inequality for positive solution of some integral equation, Ann. Mat. Pura Appl., (4) 191 (2012), 503-528.doi: 10.1007/s10231-011-0193-2.


    J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.doi: 10.1137/060676854.


    J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223.doi: 10.1016/j.anihpc.2012.07.005.


    V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.doi: 10.1007/s00285-003-0210-1.


    K. Kawasaki and N. Shigesada, An integrodifference model for biological invasions in a periodically fragmented environment, Japan J. Indust. Appl. Math., 24 (2007), 3-15.doi: 10.1007/BF03167504.


    W. T. Li, J. W. Sun and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Anal., 74 (2011), 3501-3509.doi: 10.1016/j.na.2011.02.034.


    W. T. Li, J. W. Sun and F. Y. Yang, Blow-up phenomena for nonlocal inhomogeneous diffusion problems, Turkish J. Math., 37 (2013), 466-482.


    P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235-1260.doi: 10.1016/j.matpur.2005.04.001.


    W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.doi: 10.1016/j.jde.2010.04.012.


    W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.doi: 10.1090/S0002-9939-2011-11011-6.

  • 加载中

Article Metrics

HTML views() PDF downloads(84) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint