Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity. (11th January 2017)

• Record Type: Journal Article
• Authors: Winkler, Michael
• Is Part Of: Nonlinearity. Volume 30:Number 2(2017:Feb.)
• Page Start: 735
• Page End: 764
• Publication Date: 2017-01-11
• Subjects: chemotaxis -- degenerate diffusion -- global existence -- infinite-time blow-up -- grow-up rate; 35B65 -- 35B40 (primary) -- 35K55 -- 92C17 -- 35Q30 -- 35Q92 (secondary); Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa565b
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: The Neumann initial-boundary value problem for the chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ), v t = Δ v − v + u, ( ⋆ ) is considered in a bounded domain Ω ⊂ R n, n ⩾ 1, with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D ( s ) decays exponentially as s → ∞ . In such situations, namely, straightforward Moser-type recursive arguments for the derivation of L ∞ estimates for u from corresponding L p bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context. This work develops an alternative approach, at its core based on a Moser-type iteration for the quantity e u, to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of ∥ u ( ⋅, t ) ∥ L ∞ ( Ω ) as t → ∞, under the assumptions that with some K 1 > 0, K 2 > 0, β − > 0 and β + ∈ ( − ∞, β − ] we have K 1 e − β − s ⩽ D ( s ) ⩽ K 2 e − β + s for all s ⩾ 0, and that the size of S relative to D can be estimated according to S ( s ) D ( s ) ⩽ K 3 e γ s for all s ⩾ 0 with some K 3 > 0 and γ ∈ [ β + − β − 2, β + 2 ) . Making use of the fact that this allows for certain superalgebraic growth of S D, as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when D ( s ) = e − β s and S ( s ) = s e − α s for all s ⩾ 0 and some positive α and β, the assumptions that n ⩾ 2 and that β > 0 a n d { α ∈ ( β 2, β ) i f n = 2, α ∈ ( β 2, β ] i f n ⩾ 3, warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic. To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller–Segel system of type ( ⋆ ) for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions.