Article Contents
Article Contents

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

• 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 $u_0$.
Mathematics Subject Classification: Primary: 35K57, 35B40.

 Citation:

•  [1] P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statistical Phys., 95 (1999), 1119-1139.doi: 10.1023/A:1004514803625. [2] P. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Rat. Mech. Anal., 150 (1999), 281-305.doi: 10.1007/s002050050189. [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), 105-136.doi: 10.1007/s002050050037. [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), 428-440.doi: 10.1016/j.jmaa.2006.09.007. [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, Adv. Differential Equations, 2 (2006), 271-291. [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), 123-134.doi: 10.1137/S0036141002407727. [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. [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), 53-60.doi: 10.1007/s11856-009-0019-8. [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), 137-156.doi: 10.1007/s00205-007-0062-8. [11] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003. [12] G. Gilboa and S. Osher, Nonlocal operators with application to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.doi: 10.1137/070698592. [13] L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, NJ, 2004. [14] L. Herraiz, Asymptotic behavior of solutions of some semilinear parabolic problems, Ann. Inst. Henri Poincare, 16 (1999), 49-105.doi: 10.1016/S0294-1449(99)80008-0. [15] L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, J. Evolution Equations, 8 (2008), 617-629.doi: 10.1007/s00028-008-0372-9. [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), 393-408. [17] S. Kamin and L. A. Peletier, Large time behavior of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146.doi: 10.1007/BF02801989. [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), 456-469.doi: 10.1016/0022-247X(87)90196-X. [19] C. Lederman and N. Wolanski, Singular perturbation in a nonlocal diffusion model, Communications in PDE, 31 (2006), 195-241.doi: 10.1080/03605300500358111. [20] A. Pazoto and J. D. Rossi, Asymptotic behavior for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155. [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), 1421-1432,doi: 10.1090/S0002-9939-2010-10612-3. [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), 162-196.doi: 10.1016/S0022-0396(03)00170-0. [23] J. Zhao, The Asymptotic Behavior of solutions of a quasilinear degenerate parabolic equation, J. Differential Equations, 102 (1993), 33-52.doi: 10.1006/jdeq.1993.1020.