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Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

Biler, Piotr; Corrias, Lucilla; Dolbeault, Jean (2011), Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis, Journal of Mathematical Biology, 63, 1, p. 1-32. http://dx.doi.org/10.1007/s00285-010-0357-5

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Type
Article accepté pour publication ou publié
Date
2011
Journal name
Journal of Mathematical Biology
Volume
63
Number
1
Publisher
Springer
Pages
1-32
Publication identifier
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Author(s)
Biler, Piotr
Corrias, Lucilla
Dolbeault, Jean cc
Abstract (EN)
In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M c . However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above M c always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M c , which is forbidden in the parabolic–elliptic case.
Subjects / Keywords
Keller–Segel model; Chemotaxis; Self-similar solution; Nonlocal parabolic equations; Critical mass; Existence; Blowup