Abstract
In this paper, we show that smooth solutions to the Dirichlet problem for the parabolic equation
with v(x, t) = g(x, t), \({x \in \partial \Omega,}\) can be approximated uniformly by solutions of nonlocal problems of the form
with \({u^{\varepsilon}(x,t)=g(x,t)}\), \({x \notin \Omega}\), as \({\varepsilon \to 0}\), for an appropriate rescaled kernel \({K_{\varepsilon}}\). In this way, we show that the usual local evolution problems with spatial dependence can be approximated by nonlocal ones. In the case of an equation in divergence form, we can obtain an approximation with symmetric kernels, that is, \({K_{\varepsilon}(x,y) = K_{\varepsilon}(y,x)}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal diffusion problems. Math. Surveys Monogr. 165. AMS (2010)
Bobaru F., Yang M., Frota Alves L., Silling S.A., Askari E., Xu J.: Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Meth. Eng. 77, 852–877 (2009)
Bodnar M., Velazquez J.J.L.: An integro-differential equation arising as a limit of individual cell-bases models. J. Differ. Equ. 222, 341–380 (2006)
Carrillo C., Fife P.: Spatial effects in discrete generation population models. J. Math. Biol. 50, 161–188 (2005)
Cortázar C., Coville J., Elgueta M., Martínez S.: A nonlocal inhomogeneous dispersal process. J. Differ. Equ. 241, 332–358 (2007)
Cortázar C., Elgueta M., Rossi J.D., Wolanski N.: Boundary fluxes for nonlocal diffusion. J. Differ. Equ. 234, 360–390 (2007)
Cortazar C., Elgueta M., Rossi J.D., Wolanski N.: How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rat. Mech. Anal. 187(1), 137–156 (2008)
Cortázar C., Elgueta M., Rossi J.D.: Nonlocal Diffusion problems that approximate the heat equation with Dirichlet boundary conditions. Israel J. Math. 170, 53–60 (2009)
Goldberg M.A.: The Derivative of a Determinant. Am. Math. Mon 79(10), 1124–1126 (1972)
Householder A.: The Theory of Matrices in Numerical Analysis. Dover, New York, NY (1964)
Fife, P.: Some nonlocal trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin vol. 129, 153–191 (2001)
Fournier N., Laurencot P.: Well-posedness of Smoluchowskis coagulation equation for a class of homogeneous kernels. J. Funct. Anal. 233, 351–379 (2006)
Hutson V., Martínez S., Mischaikow K., Vickers G.T.: The evolutions of dispersal. J. Math. Biol. 47, 483–517 (2003)
Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co. Pte. Ltd., (1996)
Silling S.A., Lehoucq R.B.: Convergence of Peridynamics to Classical Elasticity Theory. J. Elast. 93, 13–37 (2008)
Sun J.W., Li W.T., Yang F.Y.: Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems. Nonlinear Anal. 74, 3501–3509 (2011)
Valdinoci E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Math. Apl. SeMA 49, 33–44 (2009)
Vazquez J.L.: Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discr. Cont. Dyn. Sys. Ser. S. 7(4), 857–885 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Alexis Molino is supported by MINECO Grant MTM2015-68210-P (Spain) and Junta de Andalucía FQM-116 (Spain). Julio D. Rossi is supported by CONICET (Argentina).
Rights and permissions
About this article
Cite this article
Molino, A., Rossi, J.D. Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence. Z. Angew. Math. Phys. 67, 41 (2016). https://doi.org/10.1007/s00033-016-0649-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-016-0649-8