Global Existence and Boundedness of Solutions to a Chemotaxis-Consumption Model with Singular Sensitivity

In this paper we study the zero-flux chemotaxis-system{ut=Δu−χ∇⋅(uv∇v)vt=Δv−f(u)v in a smooth and bounded domain Ω of R2, with χ> 0 and f∈ C1(R) essentially behaving like uβ, 0 < β< 1. Precisely for χ< 1 and any sufficiently regular initial data u(x, 0) ≥ 0 and v(x, 0) > 0 on Ω¯ , we show the existence of global classical solutions. Moreover, if additionally m: = ∫ Ωu(x, 0) dx is sufficiently small, then also their boundedness is achieved.