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Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems
April  2015, 35(4): 1391-1407. doi: 10.3934/dcds.2015.35.1391

## Asymptotic behavior for a nonlocal diffusion equation on the half line

 1 Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile, Chile 2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain, Spain

Received  June 2013 Revised  March 2014 Published  November 2014

We study the large time behavior of solutions to a nonlocal diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In the far-field scale, $\xi_1\le xt^{-1/2}\le \xi_2$ with $\xi_1,\xi_2>0$, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence $tu(x,t)$ is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor $t^{3/2}$, it converges to a multiple of the unique stationary solution of the problem that behaves as $x$ at infinity. The precise proportionality factor is obtained through a matching procedure with the far-field limit. Finally, in the very far-field, $x\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is proved to be of order $o(t^{-1})$.
Citation: Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391
##### References:
 [1] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [3] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [4] C. Brändle, E. Chasseigne and R. Ferreira, Unbounded solutions of the nonlocal heat equation, Commun. Pure Appl. Anal., 10 (2011), 1663-1686. doi: 10.3934/cpaa.2011.10.1663. [5] C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol., 50 (2005), 161-188. doi: 10.1007/s00285-004-0284-4. [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [7] C. Cortázar, M. Elgueta, F. Quirós and N. Wolanski, Asymptotic behavior for a nonlocal diffusion equation in domains with holes, Arch. Ration. Mech. Anal., 205 (2012), 673-697. doi: 10.1007/s00205-012-0519-2. [8] C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [9] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions, (French) [Moments, Dirac deltas and expansion of functions], C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698. [10] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, (2003), 153-191. [11] G. Gilboa and S. Osher, Nonlocal operators with application to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [12] L. A. Herraiz, A nonlinear parabolic problem in an exterior domain, J. Differential Equations, 142 (1998), 371-412. doi: 10.1006/jdeq.1997.3358. [13] L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, J. Evol. Equ., 8 (2008), 617-629. doi: 10.1007/s00028-008-0372-9. [14] A. Wintner, On a class of fourier transforms, Amer. J. Math., 58 (1936), 45-90. doi: 10.2307/2371058.

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##### References:
 [1] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [3] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [4] C. Brändle, E. Chasseigne and R. Ferreira, Unbounded solutions of the nonlocal heat equation, Commun. Pure Appl. Anal., 10 (2011), 1663-1686. doi: 10.3934/cpaa.2011.10.1663. [5] C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol., 50 (2005), 161-188. doi: 10.1007/s00285-004-0284-4. [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [7] C. Cortázar, M. Elgueta, F. Quirós and N. Wolanski, Asymptotic behavior for a nonlocal diffusion equation in domains with holes, Arch. Ration. Mech. Anal., 205 (2012), 673-697. doi: 10.1007/s00205-012-0519-2. [8] C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [9] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions, (French) [Moments, Dirac deltas and expansion of functions], C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698. [10] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, (2003), 153-191. [11] G. Gilboa and S. Osher, Nonlocal operators with application to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [12] L. A. Herraiz, A nonlinear parabolic problem in an exterior domain, J. Differential Equations, 142 (1998), 371-412. doi: 10.1006/jdeq.1997.3358. [13] L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, J. Evol. Equ., 8 (2008), 617-629. doi: 10.1007/s00028-008-0372-9. [14] A. Wintner, On a class of fourier transforms, Amer. J. Math., 58 (1936), 45-90. doi: 10.2307/2371058.
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