Global existence and boundedness of classical solutions for a chemotaxis model with logistic source

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We consider 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 bounded domain Ω⊂Rn, n⩾1, with smooth boundary and function f is assumed to generalize the logistic source:f(u)=au−bu2,u⩾0, with a>0,b>0. Moreover, χ(s) and g(s) are nonnegative smooth functions and satisfy:χ(s)⩽ϱ(1+ϑs)k,s⩾0, with some ϱ>0,ϑ>0 and k>1,g(s)⩽h0(1+hs)δ,s⩾0,withh0>0,h⩾0,δ⩾0. We prove that for all positive values of ϱ, a and b, classical solutions to the above system are uniformly-in-time bounded. This result extends a recent result by C. Mu, L. Wang, P. Zheng and Q. Zhang (2013) [13], which asserts the global existence and boundedness of classical solutions on condition that 0⩽a<2b and ϱ be sufficiently small.