Large-time behavior of a parabolic-parabolic chemotaxis modelwith logarithmicsensitivity in one dimension

This paper deals with the chemotaxis system \ arrayll uₜ =D uₓₓ-χ [u(ln v)ₓ]ₓ, & x∈ (0, 1), \ t>0,\\ vₜ =ε vₓₓ +uv-μ v, & x∈ (0, 1), \ t>0, array . under Neumann boundary condition, where χ0, 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, thesystem possesses a unique global smooth solution that enjoysexponential convergence properties in L^∞(Ω) as timegoes to infinity, which depend on the sign of μ-ū₀, whereū₀ :=∫₀¹ u₀ dx. Moreover, we prove that the constantpair (μ, (μ/λ)D/χ) (whereλ>0 is an arbitrary constant) is the only positivestationary solution. The biological implications of our results willbe given in the paper.