Artículo

Artículo de Acceso Abierto. Puede ser descargado en su versión final
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments. © 2007 American Mathematical Society.

Registro:

Documento: Artículo
Título:Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
Autor:Bogoya, M.; Ferreira, R.; Rossi, J.D.
Filiación:Depto. de Matemática, Univ. Católica de Chile, Santiago, Chile
Depto. de Matemática, Univ. Nacional de Colombia, Bogotá, Colombia
Depto. de Matemática, U. Carlos III, 28911, Leganés, Spain
IMAFF, CSIC, Serrano 117, Madrid, Spain
Depto. Matematica, FCEyN, UBA, Buenos Aires, Argentina
Palabras clave:Neumann boundary conditions; Nonlocal diffusion
Año:2007
Volumen:135
Número:12
Página de inicio:3837
Página de fin:3846
DOI: http://dx.doi.org/10.1090/S0002-9939-07-09205-2
Handle:http://hdl.handle.net/20.500.12110/paper_00029939_v135_n12_p3837_Bogoya
Título revista:Proceedings of the American Mathematical Society
Título revista abreviado:Proc. Am. Math. Soc.
ISSN:00029939
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00029939_v135_n12_p3837_Bogoya.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v135_n12_p3837_Bogoya

Referencias:

  • Aronson, D.G., The porous medium equation (1986) Lecture Notes in Math., 1224. , in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio eds., Springer Verlag, MR877986 88a:35130
  • Bates, P., Chen, F., Spectral analysis and multidimensional stability of travelling waves for nonlocal Allen-Cahn equation (2002) J. Math. Anal. Appl., 273, pp. 45-57. , MR1933014 2003h:35104
  • Bates, P., Chmaj, A., An integrodifferential model for phase transitions: Stationary solutions in higher dimensions (1999) J. Statistical Phys., 95, pp. 1119-1139. , MR1712445 2000j:82020
  • Bates, P., Chmaj, A., A discrete convolution model for phase transitions (1999) Arch. Rat. Mech. Anal., 150, pp. 281-305. , MR1741258 2001c:82026
  • Bates, P., Fife, P., Ren, X., Wang, X., Travelling waves in a convolution model for phase transitions (1997) Arch. Rat. Mech. Anal., 138, pp. 105-136. , MR1463804 98f:45004
  • Chen, X., Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations (1997) Adv. Differential Equations, 2, pp. 125-160. , MR1424765 98f:35069
  • Cortazar, C., Elgueta, M., Rossi, J.D., A non-local diffusion equation whose solutions develop a free boundary (2005) Annales Henri Poincaré, 6 (2), pp. 269-281. , MR2136191 2006i:35379
  • Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions (2003) Trends in Nonlinear Analysis, pp. 153-191. , Springer, Berlin, MR1999098 2004h:35100
  • Lederman, C., Wolanski, N., Singular perturbation in a nonlocal diffusion model (2006) Comm. Part. Differential Equations, 31 (2), pp. 195-241. , MR2209752 2007e:35166
  • Vazquez, J.L., An introduction to the mathematical theory of the porous medium equation (1992) Shape Optimization and Free Boundaries, pp. 347-389. , in, M. C. Delfour ed., Dordrecht, Boston and Leiden, MR1260981 95b:35101
  • Wang, X., Metastability and stability of patterns in a convolution model for phase transitions (2002) J. Differential Equations, 183, pp. 434-461. , MR1919786 2003f:35157

Citas:

---------- APA ----------
Bogoya, M., Ferreira, R. & Rossi, J.D. (2007) . Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models. Proceedings of the American Mathematical Society, 135(12), 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2
---------- CHICAGO ----------
Bogoya, M., Ferreira, R., Rossi, J.D. "Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models" . Proceedings of the American Mathematical Society 135, no. 12 (2007) : 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2
---------- MLA ----------
Bogoya, M., Ferreira, R., Rossi, J.D. "Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models" . Proceedings of the American Mathematical Society, vol. 135, no. 12, 2007, pp. 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2
---------- VANCOUVER ----------
Bogoya, M., Ferreira, R., Rossi, J.D. Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models. Proc. Am. Math. Soc. 2007;135(12):3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2