Global existence and boundedness of classical solutions to a parabolic–parabolic chemotaxis system

This paper deals with the global existence and boundedness of solutions for the chemotaxis system {ut=Δu−∇⋅(uχ(v)∇v)+f(u),x∈Ω,t>0,vt=Δv−v+ug(u),x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RnΩ⊂Rn, with non-negative initial data u0∈C0(Ω¯) and v0∈W1,l(Ω)v0∈W1,l(Ω) (for some l>nl>n). The functions χχ and ff are assumed to generalize the chemotactic sensitivity function χ(s)=χ0(1+βs)2,s≥0,withβ≥0,χ0>0 and the logistic source f(s)=as−bs2,s≥0,witha>0,b>0, respectively. Here g(s)g(s) with s≥0s≥0 is a non-negative function.It is proved that the corresponding initial–boundary value problem possesses a unique global classical solution that is uniformly bounded if some technical conditions are fulfilled.