Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production. Issue 14 (25th October 2020) - Viglialoro, Giuseppe; Woolley, Thomas E.
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.