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Linear non-local diffusion problems in metric measure spaces

Part of: Integral, integro-differential, and pseudodifferential operators Qualitative behavior

Published online by Cambridge University Press:  19 July 2016

Aníbal Rodríguez-Bernal  and
Silvia Sastre-Gómez
Aníbal Rodríguez-Bernal
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain ([email protected]) and, Instituto de Ciencias Matemáticas, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13–15, 28049, Madrid, Spain
Silvia Sastre-Gómez
Affiliation:
School of Mathematical Sciences, Western Gateway Building, Western Road, University College Cork, Ireland ([email protected])
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Extract

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

Keywords

non-local diffusionlinear problemmetric measure spaces

MSC classification

Secondary: 45M05: Asymptotics 45M20: Positive solutions 47G10: Integral operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 4 , August 2016 , pp. 833 - 863
Copyright
Copyright © Royal Society of Edinburgh 2016 

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