A local/nonlocal diffusion model

ABSTRACT

In this paper, we study some qualitative properties for solutions to an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains. The coupling takes place at the interface between these two domains in such a way that the resulting evolution problem is the gradient flow of an energy functional. We prove existence and uniqueness results, as well as that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Besides, we show a suitable way to recover the heat equation at the whole domain from taking the limit at the nonlocal rescaled kernel. Finally, we propose a brief discussion about the extension of the problem to higher dimensions.

Keywords:

• Nonlocal diffusion
• heat equation
• asymptotic behavior

Mathematics Subject Classifications:

• 35K55
• 35B40
• 35A05

Acknowledgments

BCS was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (Capes) – No 88887369814/2019-00.

JDR is partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by the Spanish project MTM2015-70227-P.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Capes Consejo Nacional de Investigaciones Científicas y Técnicas and Ministerio de Ciencia, Tecnología e Innovación Productiva.