Boundary fluxes for non-local diffusion

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

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle... View more
  • References (14)
    14 references, page 1 of 2

    [1] 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).

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

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

    [4] M. Chleb´ık and M. Fila. Some recent results on blow-up on the boundary for the heat equation. Evolution equations: existence, regularity and singularities (Warsaw, 1998), 61-71, Banach Center Publ., 52, Polish Acad. Sci., Warsaw, 2000.

    [5] 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).

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

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

    [8] M. Fila and J. Filo. Blow-up on the boundary: a survey. Singularities and differential equations (Warsaw, 1993), 67-78, Banach Center Publ., 33, Polish Acad. Sci., Warsaw, 1996.

    [9] V. A. Galaktionov and J. L. Va´zquez. The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst. Vol. 8 (2) (2002), 399-433, Current developments in partial differential equations (Temuco, 1999).

    [10] B. Hu and H. M. Yin. it The profile near blowup time for solution of the heat equation with a nonlinear boundary condition. Trans. Amer. Math. Soc. 346 (1994), no. 1, 117-135.

  • Metrics
Share - Bookmark