Archive for Rational Mechanics and Analysis

, Volume 187, Issue 1, pp 137–156 | Cite as

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

  • Carmen Cortazar
  • Manuel Elgueta
  • Julio D. Rossi
  • Noemi Wolanski


We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.


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  1. 1.
    Bates P., Chmaj A. (1999). An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Statist. Phys. 95, 1119–1139MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bates P., Chmaj A. (1999). A discrete convolution model for phase transitions. Arch. Ration Mech. Anal. 150, 281–305MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bates P., Fife P., Ren X., Wang X. (1997). Traveling waves in a convolution model for phase transitions. Arch. Ration Mech. Anal. 138, 105–136MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bates P., Han J. (2005). The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation. J. Math. Anal. Appl. 311, 289–312MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bates P., Han J. (2005). The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Differential Equations. 212, 235–277MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Carrillo C., Fife P. (2005). Spatial effects in discrete generation population models. J. Math. Biol. 50, 161–188MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cortazar C., Elgueta M., Rossi J.D. (2005). A nonlocal diffusion equation whose solutions develop a free boundary. Ann. Henri Poincaré 6, 269–281MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N.: Boundary fluxes for nonlocal diffusion. To appear in J. Differential Equations Google Scholar
  9. 9.
    Chen X. (1997). Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations 2, 125–160MATHMathSciNetGoogle Scholar
  10. 10.
    Fife, P.: Some nonclassical trends in parabolic and parabolic-like evolutions. In: Trends in nonlinear analysis. Springer, Berlin, 2003Google Scholar
  11. 11.
    Friedman A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJMATHGoogle Scholar
  12. 12.
    Lederman C., Wolanski N. (2006). Singular perturbation in a nonlocal diffusion problem. Comm. Partial Differential Equations. 31, 195–241MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Wang X. (2002). Metastability and stability of patterns in a convolution model for phase transitions. J. Differential Equations. 183: 434–461MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Zhang L. (2004). Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. J. Differential Equations 197: 162–196MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Carmen Cortazar
    • 1
  • Manuel Elgueta
    • 1
  • Julio D. Rossi
    • 2
  • Noemi Wolanski
    • 2
  1. 1.Departamento de MatemáticaUniversidad Catolica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaFCEyN, UBA, Ciudad Universitaria, Pab IBuenos AiresArgentina

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