Solvability of a Keller–Segel system with signal-dependent sensitivity and essentially sublinear production

In this paper, we consider the zero-flux chemotaxis system ut=Δu−∇⋅(uχ(v)∇v) in Ω×(0,∞),0=Δv−v+g(u) in Ω×(0,∞), in a smooth and bounded domain Ω of R2. The chemotactic sensitivity χ is a general nonnegative function from C1((0,∞)) while g, the production of the chemical signal v, belongs to C1([0,∞)) and satisfies λ1≤g(s)≤λ2(1+s)β, for all s≥ 0, 0≤β<1 and 0<λ1≤λ2. 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.