Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production. Issue 14 (25th October 2020)

• Record Type: Journal Article
• Authors: Viglialoro, Giuseppe; Woolley, Thomas E.
• Is Part Of: Applicable analysis. Volume 99:Issue 14(2020)
• Page Start: 2507
• Page End: 2525
• Publication Date: 2020-10-25
• Subjects: Nonlinear parabolic systems -- chemotaxis -- global existence -- boundedness; 35A01 -- 35K55 -- 35Q92 -- 92C17; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2019.1569227
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this paper, we consider the zero-flux chemotaxis system u t = Δ u − ∇ ⋅ ( u χ ( v ) ∇ v ) in Ω × ( 0, ∞ ), 0 = Δ v − v + g ( u ) in Ω × ( 0, ∞ ), in a smooth and bounded domain Ω of R 2 . The chemotactic sensitivity χ is a general nonnegative function from C 1 ( ( 0, ∞ ) ) while g, the production of the chemical signal v, belongs to C 1 ( [ 0, ∞ ) ) and satisfies λ 1 ≤ g ( s ) ≤ λ 2 ( 1 + s ) β, for all s ≥ 0, 0 ≤ β < 1 and 0 < λ 1 ≤ λ 2 . It is established that no chemotactic collapse for the cell distribution u occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data u ( x, 0 ) emanates a unique pair of global and uniformly bounded functions ( u, v ) which classically solve the corresponding initial boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.