Artículo Original
Publicado 2012-11-28
Palabras clave
- difusión no local,
- condiciones de Neumann,
- semidiscretización,
- explosión
Cómo citar
Bogoya, M., & Forero, A. (2012). Modelo semidiscreto para una ecuación de difusión no local con fuente. Revista Integración, Temas De matemáticas, 30(2), 107–120. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2900
Resumen
Estudiamos el modelo semidiscreto para un problema de difusión no local con fuente (ui) ′ (t) = N j=−N hJ h(i − j) uj(t) – N j=−N hJ h(i − j) ui(t) + f(ui(t)), con dato inicial ui(0) = u0(xi) > 0. Probamos la existencia y unicidad de las soluciones. Además, se demuestra que las soluciones del problema discreto convergen a las del continuo cuando el parámetro de la malla va a cero. Analizamos el fenómeno de explosión de las soluciones. Para algunas fuentes f se obtiene la razón de explosión. Finalmente se presentan algunos experimentos numéricos.
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Referencias
[1] Aronson D.G., “The porous medium equation”, Nonlinear Diffusion Problems, Lecture Notes in Math. 1224 (1986), 1–46.
[2] Bates P.W., Fife P.C., Ren X. and Wang X., “Travelling waves in a convolution model for phase transitions”, Arch. Ration. Mech. Anal. 138 (1997), no. 2, 105–136.
[3] Bogoya M., “Blow up for a nonlocal nonlinear diffusion equation with source”, Rev. Colombiana Mat. 46 (2012), no. 1, 1–13.
[4] Bogoya M., Ferreira R. and Rossi J.D., “Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models”, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3837–3846.
[5] Chen X., “Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations”. Adv. Differential Equations 2 (1997), no. 1, 125–160.
[6] Cortázar C., Elgueta M. and Rossi J.D., “A non-local diffusion equation whose solutions develop a free boundary”, Ann. Henri Poincaré 6 (2005), no. 2, 269–281.
[7] Fife P., “Some nonclassical trends in parabolic and parabolic-like evolutions”, Trends in nonlinear analysis. 153–191, Springer, Berlin, 2003.
[8] Groisman P. and Rossi J.D., “Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions”, J. Comput. Appl. Math. 135 (2001), no. 1, 135–155.
[9] Galaktionov V.A. and Vázquez J.L., “The problem of blow-up in nonlinear parabolic equations”, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 399–433.
[10] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P. and Mikhailov A.P., Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987 (in Russian). English transl.: Walter de Gruyter and Co., Berlin, 1995.
[11] Vázquez J.L., “An introduction to the mathematical theory of the porous medium equation”, Shape optimization and free boundaries(Montreal, PQ, 1990), 347–389, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, Kluwer Acad. Publ., Dordrecht, 1992.
[12] Wang X., “Metastability and stability of patterns in a convolution model for phase transitions”, J. Differential Equations 183 (2002), 434–461.
[2] Bates P.W., Fife P.C., Ren X. and Wang X., “Travelling waves in a convolution model for phase transitions”, Arch. Ration. Mech. Anal. 138 (1997), no. 2, 105–136.
[3] Bogoya M., “Blow up for a nonlocal nonlinear diffusion equation with source”, Rev. Colombiana Mat. 46 (2012), no. 1, 1–13.
[4] Bogoya M., Ferreira R. and Rossi J.D., “Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models”, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3837–3846.
[5] Chen X., “Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations”. Adv. Differential Equations 2 (1997), no. 1, 125–160.
[6] Cortázar C., Elgueta M. and Rossi J.D., “A non-local diffusion equation whose solutions develop a free boundary”, Ann. Henri Poincaré 6 (2005), no. 2, 269–281.
[7] Fife P., “Some nonclassical trends in parabolic and parabolic-like evolutions”, Trends in nonlinear analysis. 153–191, Springer, Berlin, 2003.
[8] Groisman P. and Rossi J.D., “Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions”, J. Comput. Appl. Math. 135 (2001), no. 1, 135–155.
[9] Galaktionov V.A. and Vázquez J.L., “The problem of blow-up in nonlinear parabolic equations”, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 399–433.
[10] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P. and Mikhailov A.P., Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987 (in Russian). English transl.: Walter de Gruyter and Co., Berlin, 1995.
[11] Vázquez J.L., “An introduction to the mathematical theory of the porous medium equation”, Shape optimization and free boundaries(Montreal, PQ, 1990), 347–389, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, Kluwer Acad. Publ., Dordrecht, 1992.
[12] Wang X., “Metastability and stability of patterns in a convolution model for phase transitions”, J. Differential Equations 183 (2002), 434–461.