# American Institute of Mathematical Sciences

June  2011, 31(2): 581-605. doi: 10.3934/dcds.2011.31.581

## Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data

 1 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina, Argentina

Received  April 2010 Revised  April 2011 Published  June 2011

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\rightarrow A>0$ as $|x|\rightarrow\infty$. One of our main goals is the study of the critical case $p=1+2/\alpha$ for $0 < \alpha < N$, left open in previous articles, for which we prove that $t^{\alpha/2}|u(x,t)-U(x,t)|\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.
Citation: Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581
##### References:
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##### References:
 [1] P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions,, J. Statistical Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625. Google Scholar [2] P. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Rat. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. Google Scholar [3] P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions,, Arch. Rat. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar [4] P. 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. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar [5] C. Carrillo and P. Fife, Spatial effects in discrete generation population models,, J. Math. Biol., 50 (2005), 161. doi: 10.1007/s00285-004-0284-4. Google Scholar [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, Adv. Differential Equations, 2 (2006), 271. Google Scholar [7] X. Chen, Y. W. Qi and M. Wang, Long time behavior of solutions to p-laplacian equation with absorption,, SIAM Jour. Math. Anal., 35 (2003), 123. doi: 10.1137/S0036141002407727. Google Scholar [8] C. Cortazar, M. Elgueta, F. Quiros and N. Wolanski, Large time behavior of the solution to the Dirichlet problem for a nonlocal diffusion equation in an exterior domain,, in preparation., (). Google Scholar [9] C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions,, Israel Journal of Mathematics., 170 (2009), 53. doi: 10.1007/s11856-009-0019-8. Google Scholar [10] 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. Rat. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8. Google Scholar [11] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, Trends in Nonlinear Analysis, (2003), 153. Google Scholar [12] G. Gilboa and S. Osher, Nonlocal operators with application to image processing,, Multiscale Model. Simul., 7 (2008), 1005. doi: 10.1137/070698592. Google Scholar [13] L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004). Google Scholar [14] L. Herraiz, Asymptotic behavior of solutions of some semilinear parabolic problems,, Ann. Inst. Henri Poincare, 16 (1999), 49. doi: 10.1016/S0294-1449(99)80008-0. Google Scholar [15] L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations,, J. Evolution Equations, 8 (2008), 617. doi: 10.1007/s00028-008-0372-9. Google Scholar [16] S. Kamin and L. A. Peletier, Large time behavior of solutions of the heat equation with absorption,, Anal. Scuola. Norm. Sup. Pisa Serie 4, 12 (1985), 393. Google Scholar [17] S. Kamin and L. A. Peletier, Large time behavior of solutions of the porous media equation with absorption,, Israel J. Math., 55 (1986), 129. doi: 10.1007/BF02801989. Google Scholar [18] S. Kamin and M. Ughi, On the behavior as $t\to\infty$ of the solutions of the Cauchy problem for certain nonlinear parabolic equations,, J. Math. Anal. Appl., 128 (1987), 456. doi: 10.1016/0022-247X(87)90196-X. Google Scholar [19] C. Lederman and N. Wolanski, Singular perturbation in a nonlocal diffusion model,, Communications in PDE, 31 (2006), 195. doi: 10.1080/03605300500358111. Google Scholar [20] A. Pazoto and J. D. Rossi, Asymptotic behavior for a semilinear nonlocal equation,, Asymptotic Analysis, 52 (2007), 143. Google Scholar [21] J. Terra and N. Wolanski, Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case,, Proc. Amer. Math. Soc., 139 (2011), 1421. doi: 10.1090/S0002-9939-2010-10612-3. Google Scholar [22] L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks,, J. Differential Equations, 197 (2004), 162. doi: 10.1016/S0022-0396(03)00170-0. Google Scholar [23] J. Zhao, The Asymptotic Behavior of solutions of a quasilinear degenerate parabolic equation,, J. Differential Equations, 102 (1993), 33. doi: 10.1006/jdeq.1993.1020. Google Scholar
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