A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data

• Published in: Nonlinear differential equations and applications Vol. 24; no. 4
• Main Authors: Lankeit, Johannes, Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2017; Springer Nature B.V
• Subjects: Analysis; Boundary value problems; Mathematics; Mathematics and Statistics; Regularity; Logarithmic sensitivity; Generalized solution; 35D99; Global existence; 92C17; Chemotaxis; 35K55
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-017-0472-8

The chemotaxis system u t = Δ u - χ ∇ · u v ∇ v , v t = Δ v - v + u , is considered in a bounded domain Ω ⊂ R n with smooth boundary, where χ > 0 . An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter χ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that χ < ∞ if n = 2 , 8 if n = 3 , n n - 2 if n ≥ 4 , for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that u ∈ L l o c 1 ( Ω ¯ × [ 0 , ∞ ) ) .