Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions

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This paper deals with the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source, (0.1)ut=Δu−χ∇⋅(uv∇v)+u(a(t,x)−b(t,x)u),x∈Ω0=Δv−μv+νu,x∈Ω∂u∂n=∂v∂n=0,x∈∂Ω,where Ω⊂RN is a smooth bounded domain, a(t,x) and b(t,x) are positive smooth functions, and χ, μ and ν are positive constants. In recent years, it has been drawn a lot of attention to the question of whether logistic kinetics prevents finite-time blow-up in various chemotaxis models. In the very recent paper (Kurt and Shen, 2021), we proved that for every given nonnegative initial function 0⁄≡u0∈C0(Ω̄) and s∈R, (0.1) has a unique globally defined classical solution (u(t,x;s,u0),v(t,x;s,u0)) with u(s,x;s,u0)=u0(x), which shows that, in any space dimensional setting, logistic kinetics prevents the occurrence of finite-time blow-up even for arbitrarily large χ. In Kurt and Shen (2021), we also proved that globally defined positive solutions of (0.1) are uniformly bounded under the assumption (0.2)ainf>μχ24if0<χ≤2μ(χ−1)ifχ>2.In this paper, we further investigate qualitative properties of globally defined positive solutions of (0.1) under the assumption (0.2). Among others, we provide some concrete estimates for ∫Ωu−p and ∫Ωuq for some p>0 and q>2N and prove that any globally defined positive solution is bounded above and below eventually by some positive constants independent of its initial functions. We prove the existence of a “rectangular” type bounded invariant set (in Lq) which eventually attracts all the globally defined positive solutions. We also prove that (0.1) has a positive entire classical solution (u∗(t,x),v∗(t,x)), which is periodic in t if a(t,x) and b(t,x) are periodic in t and is independent of t if a(t,x) and b(t,x) are independent of t.