A generalization of Alon–Aydinian–Huang theorem

Let $S_{n,k}^m$ be the collection of sets of real numbers of size n, in which every subset of size larger than k has a sum less than m, where $n\ge k+1$, and m is some real number. Denote by $a_{n,k}^m$ the maximum number of nonempty subsets of a set in $S_{n,k}^m$ with a sum at least m. In particular, when $m=0$, Alon, Aydinian, Huang ((2014) [1]) proved that $a_{n,k}^0={\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)$, where two technical proofs, based on a weighted version of Hall's theorem and an extension of the nonuniform Erd?s–Ko–Rado theorem, were presented. In this note, we extend their elegant result from $m=0$ to any real number m, and show that $a_{n,k}^m=\{\begin{array}{cc}{\sum }_{i=0}^{k?1}\left(\begin{array}{c}n?1\\ i\end{array}\right)\hfill & \text{ if }m\ge 0\hfill \\ {\sum }_{i=1}^k\left(\begin{array}{c}n\\ i\end{array}\right)\hfill & \text{ if }m<0\hfill \end{array}$. Our proof is obtained by exploring the recurrence relation and initial conditions of $a_{n,k}^m$.     

Large time behavior in a fractional chemotaxis–Navier–Stokes system with logistic source

This paper deals with a coupled chemotaxis–Navier–Stokes system with logistic source and a fractional diffusion of order $\alpha \in (\frac{1}{2},1)$ $\left\{\begin{array}{c}{n}_t+u\cdot \nabla n=-{(-\Delta )}^{\alpha }n-\chi \nabla \cdot (n\nabla c)+an-b{n}^2,\hfill \\ c_t+u\cdot \nabla c=\Delta c-nc,\hfill \\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \varphi +f,\nabla \cdot u=0\hfill \end{array}\right.$on three dimensional periodic torus ${\mathbb{T}}^3$. Since there is no classical solution in the three-dimensional full Navier–Stokes equations, our main purpose of this paper is to investigate the global existence of weak solutions to the above system in the case of a weaker diffusion, and after some waiting time, the weak solutions in fact become smooth and converge to the semi-trivial steady state $(\frac{a}{b},0,0).$     

Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system $\left\{\begin{array}{cc}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\nabla \cdot (G\left(u\right)\chi \left(v\right)\nabla v)+\nabla \cdot (H\left(u\right)\xi \left(w\right)\nabla w),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_1\Delta v+\alpha u-\beta v,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=d_2\Delta w+\gamma u-\delta w,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 1)$ is a bounded domain with smooth boundary, $d_1,d_2,\alpha ,\beta ,\gamma ,\delta >0$ are constants. Also, $D,G,H\in C^2\left([0,\infty )\right)$ fulfill that $a_0{(s+1)}^{m-1}\le D\left(s\right)\le a_1{(s+1)}^{m-1}$ with $a_0,a_1>0$ and $m\in \mathbb{R}$; $G\left(0\right)=0$, $0\le G\left(s\right)\le b_0{(s+1)}^{q-1}$ with $b_0>0$ and $q<min\{2,m+1\}$; $H\left(0\right)=0$, $0\le H\left(s\right)\le c_0{(s+1)}^{r-1}$ with $c_0>0$ and $r<min\{2,m+1\}$, and $\chi ,\xi$ satisfy that $0\le \chi \left(s\right)\le \frac{{\chi }_0}{s^{k_1}}$ with ${\chi }_0>0$ and $k_1>1$; $0\le \xi \left(s\right)\le \frac{{\xi }_0}{s^{k_2}}$ with ${\xi }_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.