Boundedness on a fully parabolic singular chemotaxis system with indirect signal production and logistic source

In this paper, we study the following quasilinear chemotaxis model with singular sensitivity and indirect signal production (0.1)ut=∇⋅(D(u)∇u)−χ∇⋅(uvk∇v)+u(a−bu),t>0,x∈Ω,vt=Δv−v+w,t>0,x∈Ω,wt=Δw−w+u,t>0,x∈Ω,under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω⊂RN, N≥3, where, χ,a,b>0, k∈(0,12)∪(12,1] and D(u) is smooth function satisfying D(u)≥(u+1)αwithα>0.By constructing suitable Lyapunov functional, we prove that if a>χ24, there exists a positive constant m∗ such that ∫Ωu≥m∗.Based on above result, the lower bound estimates for w and v have been established. When b is large enough and α>max{0,1−4N}, we further prove that the system has a global bounded classical solution. Finally, it is shown that the solution exponentially converges to the constant stationary solution (ab,ab,ab) as t→∞.