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

Abstract

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→∞.