Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension

This paper deals with the chemotaxis system {ut=Du xx-X[u(In ν)x]x, νt= ενxx+uν-μν, x∈(0,1), t>0, x∈(0,1), t>0 under Neumann boundary condition, where χ < 0, D > 0, ε > 0 and μ > 0 are constants. It is shown that for any sufficiently smooth initial data (u0; v0) fulfilling u0 ≥ 0, u0 ≢ 0 and v0 > 0, the system possesses a unique global smooth solution that enjoys exponential convergence properties in L∞(Ω) as time goes to infinity which depend on the sign of μ-ū0, where ū0:=∫01u0dx. Moreover, we prove that the constant pair (μ,(μ)D/χ (where > 0 is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.