Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis
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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.
KeywordsKeller–Segel model Chemotaxis Self-similar solution Nonlocal parabolic equations Critical mass Existence Blowup
Mathematics Subject Classification (2000)35B30 35K40 35K57 35J60
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- Biler P (2006) A note on the paper of Y. Naito: “Asymptotically self-similar solutions for the parabolic system modelling chemotaxis”. In: Self-similar solutions of nonlinear PDE, vol 74. Banach Center Publications, Polish Academy of Sciences, Warsaw, pp 33–40Google Scholar
- Naito Y (2006) Asymptotically self-similar solutions for the parabolic system modelling chemotaxis. In: Self-similar solutions of nonlinear PDE, vol 74. Banach Center Publications, Polish Academy of Sciences, Warsaw, pp 149–160Google Scholar