American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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