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 and Continuous Dynamical Systems, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581
References:
[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.

show all references

References:
[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.

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