Research and Innovation Articles
A semidiscrete model for a non-local diffusion equation with a source
Published 2012-11-28
Keywords
- nonlocal diffusion,
- Neumann boundary conditions,
- semidiscretization,
- blow-up
How to Cite
Bogoya, M., & Forero, A. (2012). A semidiscrete model for a non-local diffusion equation with a source. Revista Integración, Temas De matemáticas, 30(2), 107–120. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2900
Abstract
We study a discrete model for a non-local diffusion problem with a source, namely (ui) ′ (t) = Nj=−N hJ h(i − j) uj(t) – N j=−N hJ h(i − j) ui(t) + f(ui(t)), with initial datum ui(0) = u0(xi) > 0. We prove the existence and uniqueness of the solution. Moreover, we show that solutions of discrete problem converge to the continuous ones when the mesh parameter goes to zero. We also study the blow-up phenomena of solutions. For some sources f, we obtain the blow-up rate. Finally, we perform some numerical experiments.
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References
[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.