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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]

This note is to make some corrections on the assumptions in Theorems 1.2, 1.3, and 1.5 in our paper [Halil Ibrahim Kurt and Wenxian Shen, Journal of Differential Equations 355 (2023) 248–295]. [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]

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]

This study investigates the influence of titanium (Ti) and magnesium (Mg) additions on aluminum (Al) alloys in order to evaluate the relationship between the structure and properties of the new alloys. The alloys obtained at elevated temperatures mainly consist of Al-2Mg-lTi, Al-2Mg-3Ti, Al-4Mg-2Ti, and Al-6Mg-2Ti alloys, as well as α and τ solid solution phases of intermetallic structures. Microstructural analyses were performed using X-ray diffraction (XRD), optical microscope, and energy dispersive spectrometry (EDS) techniques. Test results show that the average grain size of the alloys decreased with the addition of Ti inclusions during the casting and solidification processes, and the smallest grain size was found to be 90 µm for the Al-6Mg-3Ti alloy. In addition, tensile properties of the Al-Mg-Ti alloys were initially improved and then worsened after the addition of higher concentrations of Ti. The highest tensile and hardness values of the alloys were Al-4Mg-2Ti (205 MPa) and Al-6Mg-3Ti (80 BHN). The primary reasons for having higher mechanical properties may be attributed to strengthening of the solid solution and refinement of the grain size and shape during the solidification process. For this study, the optimum concentrations of Ti and Mg added to the Al alloys were 4 and 2 wt.%, respectively. This study may be useful for field researchers to develop new classes of Al alloys for various industrial applications. [ABSTRACT FROM AUTHOR]

In this study, effects of friction stir processing (FSP) parameters and hybrid ratio on the ultimate tensile strength (UTS) of aluminum (Al) matrix (5083) hybrid composites (AMHCs) were investigated in detail. In order to estimate the ultimate tensile strength of the prepared composites, a mathematical formulation was derived, and test results were compared with those of the model. The effects of factors influencing strength, such as tool rotational and traverse speeds, and volume fractions of, carbon nanotube (CNT), aluminum oxide (Al 2 O 3 ), graphite (Gr), silicon carbide (SiC) and zirconium oxide (ZrO 2 ) were also studied using the proposed formulation. Test results showed that the ultimate tensile strength of these composites significantly increased with increasing CNT, tool rotational and traverse speeds. In addition, the effects of complex reinforcements with different volume fractions on the 5083 AMHCs were examined. A maximum tensile strength for the hybrid composite was found at the inclusion ratio of 10% Gr + 5% ZrO 2 . This study may open up new possibilities to developing highly strength Al alloys for various industrial applications. [ABSTRACT FROM AUTHOR]

In the current study, aluminum-magnesium-titanium alloys were manufactured with sand casting method and the effects of titanium and magnesium on the ductility, porosity, hardness and density of these alloys were investigated. The influences of these elements were also studied using multi-layer perceptron neural network approach. Regression models were advanced to check both the performance and the reliability of the proposed neural network model. It was seen that linear correlation of all output values is highest than 90%. It was also observed that Mg has a greater effect than Al and Ti on the hardness and porosity values, whereas Al has more sensitivity on the ductility of alloys. [ABSTRACT FROM AUTHOR]

The effects of temperature, time, and the additions of magnesium and copper on the wetting behavior of Al/TiC are studied theoretically. Mathematical formula is presented in explicit form. The effect of each variable is investigated by using the obtained equation. It is observed that the time and temperature have a stronger effect on the wetting of TiC in comparison to other input parameters. The proposed model shows good agreement with test results and can be used to find the wetting behavior of Al/TiC. The findings led to a new insight of the wetting process of TiC. [ABSTRACT FROM AUTHOR]