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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 this work, the Al–xMg–MgO composites were fabricated via stir casting method. The different amounts of Mg (1, 3 and 5 wt%) and MgO (0.5, 1.0 and 1.5 wt%) were added to the aluminum matrix, and the composites were casted at various temperatures of 700, 800 and 900 °C. The effects of the casting temperature (°) and the Mg (wt%) and MgO (wt%) ratios on the microstructure, physical and mechanical properties of the composites were investigated. The scanning electron microscope with electron-dispersive spectrum, density (g/cm3), porosity (%), hardness (HR-15T) and compression (MPa) test analyses were conducted on the composites. The results were also analyzed with the Taguchi experiment design method with ANOVA. The properties of the composites changed depending on the process parameters and ratios of Mg element and MgO particles. The minimum density and porosity contents were obtained as 2.62 g/cm3 and 0.6%, respectively. The increase in the casting temperature contributed to the wettability and so the properties of the samples to some extent. The additions of Mg and MgO improved the hardness (72.8 HR-15T) and compression strength (646.4 MPa) of the composites. It can be said that the main strengthening mechanisms of the Al–xMg–xMgO composites for this study were the solid solution and dispersion strengthening mechanisms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + u (a 1 − b 1 u − c 1 v) , x ∈ Ω v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + v (a 2 − b 2 v − c 2 u) , x ∈ Ω 0 = Δ w − μ w + ν u + λ v , x ∈ Ω ∂ u ∂ n = ∂ v ∂ n = ∂ w ∂ n = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, and χ i , a i , b i , c i (i = 1 , 2) and μ , ν , λ are positive constants. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove that for any given nonnegative initial data u 0 , v 0 ∈ C 0 (Ω ¯) with u 0 + v 0 ≢ 0 , (0.1) has a unique globally defined classical solution (u (t , x ; u 0 , v 0) , v (t , x ; u 0 , v 0) , w (t , x ; u 0 , v 0)) with u (0 , x ; u 0 , v 0) = u 0 (x) and v (0 , x ; u 0 , v 0) = v 0 (x) in any space dimensional setting with any positive constants χ i , a i , b i , c i (i = 1 , 2) and μ , ν , λ. Moreover, we prove that there is χ ⁎ (μ , χ 1 , χ 2) > 0 satisfying χ ⁎ (μ , χ 1 , χ 2) = { μ χ 2 4 if 0 < χ < 2 μ (χ − 1) if χ ≥ 2 , when χ 1 = χ 2 : = χ and χ ⁎ (μ , χ 1 , χ 2) ≤ min ⁡ { μ χ 2 + μ (χ 1 − χ 2) 2 4 , μ χ 1 + μ (χ 2 − χ 1) 2 4 } , when χ 1 ≠ χ 2 such that the condition min ⁡ { a 1 , a 2 } > χ ⁎ (μ , χ 1 , χ 2) implies lim sup t → ∞ ‖ u (t , ⋅ ; u 0 , v 0) + v (t , ⋅ ; u 0 , v 0) ‖ ∞ ≤ M ⁎ and lim inf t → ∞ inf x ∈ Ω ⁡ (u (t , x , u 0 , v 0) + v (t , x ; u 0 , v 0)) ≥ m ⁎ for some positive constants M ⁎ , m ⁎ independent of u 0 , v 0 , the latter is referred to as combined pointwise persistence. [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]