Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:

$\{\begin{array}{ccc}\hfill & u_t=\mathrm{?}?(\mathrm{?}u?\chi u\mathrm{?}v)+u?\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^2$ with $\chi ,\mu >0$, nonnegative initial data $u_0$, $v_0$, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any $\chi ,\mu >0$. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists $C=C(u_0,v_0,\mathrm{\Omega })>0$ such that

${6u(?,t)6}_{L^{\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\chi K(\chi ,\mu )N(\chi ,\mu )]$

and

${6v(?,t)6}_{W^{1,\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\frac{{\chi }^{\frac{8}{3}}}{\mu }{K}^{\frac{8}{3}}(\chi ,\mu )]=:CN(\chi ,\mu )$

uniformly on $[0,\infty )$, where

$K(\chi ,\mu )=M(\chi ,\mu )E(\chi ,\mu ),M(\chi ,\mu )=1+\frac{1}{\mu }+\sqrt{\chi }(1+\frac{1}{{\mu }^2})$

and

$E(\chi ,\mu )=\exp ?\left[\frac{\chi C_{GN}^2}{2\min ?\{1,\frac{2}{\chi }\}}\right(\frac{4}{\mu }{6u_{0}6}_{L^1(\mathrm{\Omega })}+\frac{13}{2{\mu }^2}|\mathrm{\Omega }|+{6\mathrm{?}{v}_{0}6}_{L^2(\mathrm{\Omega })}^2\left)\right].$

We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at $\mu =0$, where the corresponding minimal model (removing the term $u?\mu u^2$ in the first equation) is widely known to possess blow-ups for large initial data.     

四甲基吡啶卟啉与核酸相互作用的稳态和时间分辨光谱

采用稳态吸收和荧光光谱、圆二色谱和皮秒时间分辨荧光光谱手段, 研究了5,10,15,20-四[4-(N-甲基吡啶)]卟啉(TMPyP4)与腺嘌呤(A)、鸟嘌呤(G)、胸腺嘧啶(T)和胞嘧啶(C)等4种碱基, 以及相应的核苷、核苷酸和单链DNA的结合能力和光谱学性质. 研究结果发现, 嘌呤与TMPyP4的结合能力比嘧啶的强. 对于某一碱基系列, 结合能力强弱顺序依次为: 碱基~核苷<核苷酸<单链DNA. 时间分辨荧光谱研究发现, 除鸟嘌呤外, 核酸和TMPyP4复合物的荧光动力学均含有快(1~2 ns)和慢(约10 ns)两个衰减过程, 它们分别是由激基复合体和环境极性对激发态TMPyP4分子的影响所致. 单链DNA能诱导TMPyP4产生诱导圆二色信号, 而单分子(碱基、核苷、核苷酸)则无此功能.     

Zeolite NaY-mediated oxidation of dyes with H2O2: unique heterogeneous non-transition metal center cleavage of H2O2 under visible light irradiation

This study investigated the visible-light catalysis mediated by zeolite NaY on the oxidation of dyes with H₂O₂. The results demonstrated that zeolite NaY acts as a sink for the electron from the photo-excited dye in the heterogeneous catalysis. Furthermore, the electron can effectively activate H₂O₂ to produce ^·OH radical that is a powerful oxidant for the oxidation of dye at room temperature. The effects of the framework topology, Si/Al ratio, and exchangeable cation of the zeolite on the oxidation of various dyes were also shown.     

单道扫描式ICP—AES法测定婴儿乳粉,代乳粉中K,Na,Ca,Mg,P,Fe,Zn,Cu,Mn和Sr

本文采用HNO_3-HClO_4湿法消解样品,用乳化剂OP解决雾化器堵塞问题,利用单道扫描式ICP-AES仪测定婴儿乳粉、代乳粉中十种元素(K、Na、Ca、Mg、F、Fe、Zn、Cu、Mn和Sr)。方法简便快速,回收率在90-105%,相对标准偏差均小于10%,适用于婴儿乳粉、代乳粉中上述元素的测定。     

On the results of Lahiri concerning uniqueness of meromorphic functions

In this paper we study the problem of meromorphic functions sharing three values with weight and obtain some theorems which improve the results given by Lahiri and others.     

A sharp form of Moser-Trudinger inequality in high dimension

Let Ω be a bounded smooth domain in Rn(n?3). This paper deals with a sharp form of Moser-Trudinger inequality. Let

     

Some Results behind Dividend Problems

We consider the basic dividend problem of the compound Poisson model with constant barrierstrategy.Some results concealed behind the dividend problem are made explicit in the present work.Differentmethods and some of which are firstly given in this paper.All these results presented certain direct relationshipbetween some important actuary variables in classical risk theory is also revealed.     

上海市电力需求的协整与误差修正分析

本文采用协整与误差修正模型技术研究了上海市电力需求的决定因素,建立了上海电力需求函数,检验了电力需求与国内生产总值、电力价格、经济结构、电力使用效率之间的长期均衡关系及短期波动关系。     

多重跳图的平面性

The infinite sequence {J^k5(G)} where Js(G) denotes the 5-jump graph of G, is planar if, and only if, G=cor(K3). For r-jump graph with r≥6, there does not exist a graph G such that the sequence {J^kr(G)} is planar.     

基于非对称损失函数的参数设计

本文针对望目特性的正态指标,在非对称的二次质量损失函数下,讨论了参数设计的可行性, 证明田口方法的稳健性设计和灵敏度设计依然行之有效.定义了调整参数,求出了使质量损失最小的数值解,并给出了参数设计的具体步骤.