Large time behavior of solution to a fully parabolic chemotaxis system with singular sensitivity and logistic source

This paper presents the large time behavior of solution to the fully parabolic chemotaxis system with singular sensitivity and logistic source ut=∇⋅(D(u)∇u)−χ∇⋅(uvκ∇v)+μu−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,with homogeneous Neumann boundary condition in a convex smooth bounded domain Ω⊂Rn, n≥2, where χ>0, μ>0 and κ∈(0,12)∪(12,1), D(u) is supposed to satisfy the following property D(u)≥(u+1)αwithα>0.One can find a positive constant m∗ such that ∫Ωu≥m∗for allt≥0. Apart from that, it is shown that the solution is globally bounded. Furthermore, it is asserted that the solution exponentially converges to the steady state (1,1) as t→∞.