How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems

Preprint English OPEN
Cortazar, C. ; Elgueta, M. ; Rossi, J. D. ; Wolanski, N. (2006)
  • Subject: Mathematics - Analysis of PDEs | 35K57, 35B40

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.
  • References (14)
    14 references, page 1 of 2

    [1] P. Bates and A. Chmaj. An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Statistical Phys., 95, 1119-1139, (1999).

    [2] P. Bates and A. Chmaj. A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal., 150, 281-305, (1999).

    [3] P. Bates, P. Fife, X. Ren and X. Wang. Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal., 138, 105-136, (1997).

    [4] P. Bates and J. Han. The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation. To appear in J. Math. Anal. Appl.

    [5] P. Bates and J. Han. The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Differential Equations, 212, 235-277, (2005).

    [6] C. Carrillo and P. Fife. Spatial effects in discrete generation population models. J. Math. Biol. 50(2), 161-188, (2005).

    [7] C. Cortazar, M. Elgueta and J. D. Rossi. A non-local diffusion equation whose solutions develop a free boundary. Ann. Henri Poincare, 6(2), 269-281, (2005).

    [8] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski. Boundary fluxes for non-local diffusion. Preprint.

    [9] X Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, 125-160, (1997).

    [10] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003.

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