publication . Preprint . 2006

Boundary fluxes for non-local diffusion

Cortazar, C.; Elgueta, M.; Rossi, J. D.; Wolanski, N.;
Open Access English
  • Published: 03 Jul 2006
Abstract
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. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition.
Subjects
free text keywords: Mathematics - Analysis of PDEs, 35K57, 35B40
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[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). [OpenAIRE]

[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. [OpenAIRE]

[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). [OpenAIRE]

[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.

[11] C. Lederman and N. Wolanski. Singular perturbation in a nonlocal diffusion problem. Preprint

[12] D. Rial and J. D. Rossi. Blow-up results and localization of blow-up points in an N -dimensional smooth domain. Duke Math. J. 88(2), 391-405, (1997). [OpenAIRE]

[13] A. Samarski, V. A. Galaktionov, S. P. Kurdyunov and A. P. Mikailov. Blow-up in quasilinear parabolic equations. Walter de Gruyter, Berlin, (1995). [OpenAIRE]

[14] X. Wang. Metaestability and stability of patterns in a convolution model for phase transitions. Preprint.

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