Finite mass solutions for a nonlocal inhomogeneous dispersal equation

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April 2015, 35(4): 1409-1419. doi: 10.3934/dcds.2015.35.1409

Finite mass solutions for a nonlocal inhomogeneous dispersal equation

Carmen Cortázar ^1, , Manuel Elgueta ^2, , Jorge García-Melián ^3, and Salomé Martínez ^4,

1.	Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22. Santiago, Chile
2.	Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Santiago
3.	Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271. La Laguna, Spain
4.	Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, silla 170 Correo 3, Santiago

Received October 2013 Revised March 2014 Published November 2014

Abstract
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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.

Keywords: integro-differential equations., nonlocal, Inhomogeneous dispersal.

Mathematics Subject Classification: 45A05, 45J0.

Citation: Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409

References:

[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.
[2]	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.
[3]	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.
[4]	C. Cortázar, M. Elgueta, S. Martínez and J. Rossi, Random walks and the porous medium equation, Rev. Un. Mat. Argentina, 50 (2009), 149-155.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	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.
[15]	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.

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References:

[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.
[2]	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.
[3]	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.
[4]	C. Cortázar, M. Elgueta, S. Martínez and J. Rossi, Random walks and the porous medium equation, Rev. Un. Mat. Argentina, 50 (2009), 149-155.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	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.
[15]	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.

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