Chemotaxis systems with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions. (February 2023) - Kurt, Halil Ibrahim; Shen, Wenxian
Abstract: This paper deals with the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source, (0.1) u t = Δ u − χ ∇ ⋅ ( u v ∇ v ) + u ( a ( t, x ) − b ( t, x ) u ), x ∈ Ω 0 = Δ v − μ v + ν u, x ∈ Ω ∂ u ∂ n = ∂ v ∂ n = 0, x ∈ ∂ Ω, where Ω ⊂ R N 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 ⁄ ≡ u 0 ∈ C 0 ( Ω ̄ ) and s ∈ R, (0.1) has a unique globally defined classical solution ( u ( t, x ; s, u 0 ), v ( t, x ; s, u 0 ) ) with u ( s, x ; s, u 0 ) = u 0 ( 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) a inf > μ χ 2 4 if 0 < χ ≤ 2 μ ( χ − 1 ) if χ > 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 ∫ Ω u − p and ∫ Ω u q for some p > 0 and q > 2 N 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 L q ) 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 .