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Published in:
2021 | OriginalPaper | Chapter
3. Basic Properties
Authors : Andreas Buttenschön, Thomas Hillen
Published in: Non-Local Cell Adhesion Models
Publisher: Springer International Publishing
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Abstract
In this chapter, we define the non-local operator \({\mathcal {K}}[u]\), and we collect some basic properties of \({\mathcal {K}}[u]\) in one spatial dimension. We prove results on integrability, continuity, regularity, positivity, and a priori estimates, and we show that \({\mathcal {K}}[u]\) is a compact operator. We analyze the corresponding spectrum of \({\mathcal {K}}\), and we use these properties to derive properties of steady-state solutions such as symmetries, regularities, and a priori estimates. We find that the non-local term \({\mathcal {K}}[u]\) acts like a non-local derivative and the term \({\mathcal {K}}[u]'\) acts like a non-local curvature, in a sense made precise later.