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:
[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.

[2]

P. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Rat. Mech. Anal., 150 (1999), 281. 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. 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. 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. 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.

[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.

[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. 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. doi: 10.1007/s00205-007-0062-8.

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, Trends in Nonlinear Analysis, (2003), 153.

[12]

G. Gilboa and S. Osher, Nonlocal operators with application to image processing,, Multiscale Model. Simul., 7 (2008), 1005. doi: 10.1137/070698592.

[13]

L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004).

[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.

[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.

[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.

[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.

[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.

[19]

C. Lederman and N. Wolanski, Singular perturbation in a nonlocal diffusion model,, Communications in PDE, 31 (2006), 195. doi: 10.1080/03605300500358111.

[20]

A. Pazoto and J. D. Rossi, Asymptotic behavior for a semilinear nonlocal equation,, Asymptotic Analysis, 52 (2007), 143.

[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.

[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.

[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.

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. 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. 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. 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. 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. 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.

[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.

[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. 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. doi: 10.1007/s00205-007-0062-8.

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, Trends in Nonlinear Analysis, (2003), 153.

[12]

G. Gilboa and S. Osher, Nonlocal operators with application to image processing,, Multiscale Model. Simul., 7 (2008), 1005. doi: 10.1137/070698592.

[13]

L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004).

[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.

[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.

[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.

[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.

[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.

[19]

C. Lederman and N. Wolanski, Singular perturbation in a nonlocal diffusion model,, Communications in PDE, 31 (2006), 195. doi: 10.1080/03605300500358111.

[20]

A. Pazoto and J. D. Rossi, Asymptotic behavior for a semilinear nonlocal equation,, Asymptotic Analysis, 52 (2007), 143.

[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.

[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.

[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.

[1]

Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks & Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749

[2]

Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013

[3]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

[4]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[5]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229

[6]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049

[7]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087

[8]

Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure & Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959

[9]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

[10]

Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155

[11]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

[12]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[13]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

[14]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41

[15]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93

[16]

Shijin Deng. Large time behavior for the IBVP of the 3-D Nishida's model. Networks & Heterogeneous Media, 2010, 5 (1) : 133-142. doi: 10.3934/nhm.2010.5.133

[17]

Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077

[18]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[19]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

[20]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]