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

In this paper, we consider the zero-flux chemotaxis system ut0=Δu−∇⋅(uχ(v)∇v) in Ω×(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