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In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: ut = uxx − χ u vvxx + u(a − bu), 0 < x < L,t > 0,0 = vxx − μv + νu, 0 < x < L,t > 0,ux(t, 0) = ux(t,L) = vx(t, 0) = vx(t,L) = 0t > 0, where χ, a, b, μ, ν are positive constants. Among others, we prove there are χ∗ > 0 and {χk∗}⊂ [χ∗,∞) (χ∗∈{χ k∗}) such that the constant solution (a b , ν μ a b ) of system is locally stable when 0 < χ < χ∗ and is unstable when χ > χ∗, and under some generic condition, for each k ≥ 1, a (local) branch of nonconstant stationary solutions of system bifurcates from (a b , ν μ a b ) when χ passes through χk∗, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions {(u(⋅; χn),v(⋅; χn))} of system with χ = χn(→∞) develops spikes at any x∗ satisfying lim infn→∞u(x∗; χ n) > a b. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a b , ν μ a b ) when χ passes through χ∗ can be extended to χ = ∞ and the stationary solutions on this global bifurcation extension are locally stable when χ ≫ 1 and develop spikes as χ →∞. [ABSTRACT FROM AUTHOR]

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. However, for several important chemotaxis models, only in the space two dimensional setting has it been shown that logistic kinetics is sufficient to enforce the global existence of classical solutions (see [K. Fujie, M. Winkler, and T. Yokota, Nonlinear Anal., 109 (2014), pp. 56--71], [J. I. Tello and W. Winkler, Comm. Partial Differential Equations, 32 (2007), pp. 849--877]). The current paper gives a confirmed answer to the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting: ut=🜂u-X🜄.(u/v🜄u) + u(a(x, t) - b(x, t)u) and 0=🜂v-𝝻v+vu under the homogeneous Neumann boundary conditions in a bounded domain Ω ᑕ Rn where X is the singular chemotaxis sensitivity coefficient, a(x, t) and b(x, t) are positive smooth functions, and 𝝻, vare positive constants. We prove that, for every given nonnegative initial data 0 ≢ u0 Ⲉ C 0 (⩃) the \= system has a unique globally defined classical solution (u(t, x; u0), v(t, x; u0)) with u(0, x; u0) = u0(x), which shows that, in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large ainf ᐳ 𝝻x²/4 when X≤2 and ainf ᐳ 𝝻 (X-1) when xᐳ2. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]