Blow-up for a Nonlocal Nonlinear Diffusion Equation with Source

We study the initial-value problem prescribing Neumann boundary conditions for a nonlocal nonlinear diffusion operator with source, in a bounded domain in $\mathbb{R}^N$ with a smooth boundary.
We prove existence, uniqueness of solutions and we give a
comparison principle for its solutions. The blow-up phenomenon is
analyzed. Finally, the blow up rate is given for some particular
sources.

Explosión para una ecuación no lineal de difusión no local con fuente MAURICIO BOGOYA1

1Universidad Nacional de Colombia, Bogotá, Colombia. Email:mbogoyal@unal.edu.co 

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Abstract

We study the initial-value problem prescribing Neumann boundary conditions for a nonlocal nonlinear diffusion operator with source, in a bounded domain in RN with a smooth boundary. We prove existence, uniqueness of solutions and we give a comparison principle for its solutions. The blow-up phenomenon is analyzed. Finally, the blow up rate is given for some particular sources.

Key words: Nonlocal diffusion, Neumann boundary conditions, Blow-up.

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2000 Mathematics Subject Classification: 35K57, 35B40.

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Resumen

Se estudia el problema de valor inicial con condiciones de Neumann para un operador no lineal de difusión no local con fuente, en un dominio acotado en RN con frontera suave. Se demuestra la existencia y unicidad de las soluciones y se da un principio de comparación para las soluciones. Se analiza el fenómeno de explosión. La razón de explosión es dada para algunas fuentes particulares.

Palabras clave: Difusión no local, condiciones de Neumann, explosión.

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Texto completo disponible en PDF

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References

[1] D. G. Aronson, The Porous Medium Equation, 'Lecture Notes in Math', 1986, Vol. 1224, Springer Verlag.

[2] M. Bogoya, 'A Nonlocal Nonlinear Diffusion Equation in Higher Space Dimensions', J. Math. Anal. Appl 344, (2008), 601-615.

[3] C. Cortázar, M. Elgueta, and J. D. Rossi, 'A Non-local Diffusion Equation whose Solutions Develop a Free Boundary', Ann. Henri Poincaré 6, 2 (2005), 269-281.

[4] K. Deng and H. A. Levine, 'The Role of Critical Exponents in Blow-Up Theorems: The Sequel', J. Math. Anal. Appl 243, (2000), 85-126.

[5] P. Fife, 'Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions', Trends in nonlinear analysis2003, (2000), 153-191.

[6] A. Friedman and B. McLeod, 'Bow-up of Positive Solutions of Semilinear Heat Equations', Indiana Univ. Math. J34, 2 (1985), 425-447.

[7] V. A. Galaktionov and J. L. Vázquez, 'The Problem of Blow-up in Nonlinear Parabolic Equations', Discrete Contin. Dyn. Syst 8, 2 (2002), 399-433.

[8] H. A. Levine, 'The Role of Critical Exponents in Blow-up Theorems', SIAM Reviews 32, (1990), 262-288.

[9] B. P., F. P., R. X., and W. X., 'Travelling Waves in a Convolution Model for Phase Transitions', Arch. Rat. Mech. Anal 138, (1997), 105-136.

[10] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995.

[11] J. L. Vazquez, 'An Introduction to the Mathematical Theory of the Porous Medium Equation', Shape optimization and free boundaries, (1992), 347-389.

(Recibido en febrero de 2011. Aceptado en enero de 2012)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv46n1a01, 
AUTHOR = {Bogoya, Mauricio}, 
TITLE = {{Blow-up for a Nonlocal Nonlinear Diffusion Equation with Source}}, 
JOURNAL = {Revista Colombiana de Matemáticas}, 
YEAR = {2012}, 
volume = {46}, 
number = {1}, 
pages = {1--13} 
}