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.
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.