Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on RN

In the current paper, we consider the following parabolic-elliptic semilinear Keller-Segel model on RN,{ut=∇⋅(∇u−χu∇v)+au−bu2,x∈RN,t>0,0=(Δ−I)v+u,x∈RN,t>0, where χ>0,a≥0,b>0 are constant real numbers and N is a positive integer. We first prove the local existence and uniqueness of classical solutions (u(x,t;u0),v(x,t;u0)) with u(x,0;u0)=u0(x) for various initial functions u0(x). Next, under some conditions on the constants a,b,χ and the dimension N, we prove the global existence and boundedness of classical solutions (u(x,t;u0),v(x,t;u0)) for given initial functions u0(x). Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants a,b,χ and the dimension N, we show that for every strictly positive initial function u0(⋅),limt→∞⁡supx∈RN⁡[|u(x,t;u0)−ab|+|v(x,t;u0)−ab|]=0, and that for every nonnegative initial function u0(⋅) with non-empty and compact support supp(u0), there are 0cup⁎(u0).