Asymptotic behavior of the solutions to the one-dimensional nonisentropic hydrodynamic model for semiconductors

In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson (or full hydrodynamic) model for semiconductors, where the energy equation with non-zero thermal conductivity coefficient are contained, is discussed. The global existence of smooth solutions for the Cauchy problem with small perturbed initial data is proved. In particular, that the solutions converge to the corresponding stationary solutions exponentially fast as t → ∞ is showed. Biography: LI Yeping(1972–), male, Associate professor, Ph.D., research direction: nonlinear partial differential equations.     

具有变号非线性项的分数阶微分方程边值问题正解的存在性

为了进一步研究非线性项的分数阶微分方程边值问题的性质,讨论了带有变号非线性项的(n-1,1)分数阶微分方程特征值问题正解的存在性,其中分数阶导数是Riemann-Liouville型。首先利用给定边值问题的Green函数,将微分方程转化为等价的积分方程,然后在非线性项f(t,x)满足Caratheodory条件(即任意选取变量x,非线性项f(t,x)为可测函数,对(0,1)区间内几乎所有t,非线性项f(t,x)为x的连续函数)下。通过构造适当的Banach空间,运用锥拉伸与锥压缩不动点定理和Leray-Schauder非线性抉择得出边值问题正解存在的充分条件。结果表明,非线性项f(t,x)中的t可以在(0,1)区间内任何点处具有奇性,同时还改变了使边值问题的解存在的特征值λ的取值范围。研究结果为现存结论的深入研究打下了基础。     

具有p-Laplacian算子的共振微分方程组解的存在性

为了研究具有非线性分数阶微分算子的微分方程共振边值问题解的存在性,引入了推广的Mawhin连续定理,通过定义合适的Banach空间及范数,给出恰当的算子,运用Mawhin连续定理的拓展,研究了具有p-Laplacian算子的分数阶共振微分方程组边值问题解的存在性。通过举例验证了所得结论的正确性。所得结论是共振边值问题现有成果的推广和一般化,对进一步研究具有一定参考价值。     

两相连续铸钢问题非稳态解的渐近性质

 在已经证明了两相连续铸钢问题非稳态解的存在性,唯一性及周期解的存在性的基础上,进一步研究了解的渐近性质:如果边界上给出周期边界条件,则当t^→+∞时初边值问题的解一致收敛到周期解.主要方法是估计未知函数的一致连续模.     

Characteristic singular integral equations with solutions having singularities of higher order on the real axis

Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter. Biography: ZHONG Shouguo (1941–), male, Professor, research direction: singular integral equations and their applications.     

有限区间上广义BBM-Burgers方程解的大时间性态

研究广义BBM-Burgers方程ut-utxx-uxx+f(u)x=0(x∈[0,1],t>0)的一般初边值问题在f严格凸、初始扰动及稳定波不必小的情形下解的大时间性态.     

无穷区间上二阶三点q-差分方程边值问题解的存在性

为了拓展非线性量子差分方程边值问题的基本理论,研究了一类无穷区间上非线性项含有一阶q-微分的二阶三点非线性q-差分方程边值问题解的存在性。首先,给出并证明了含有无穷限广义积分的二重q-积分的交换积分次序公式;其次,计算出了无穷区间上二阶三点线性q-差分方程边值问题的Green函数,并研究了Green函数的性质;再次,在抽象空间上构造积分算子,然后运用Leray-Schauder连续定理,获得了无穷区间上二阶三点非线性q-差分方程边值问题解的存在性结果;最后给出实例。实例验证表明所得结果是正确的。研究结果对量子微积分的发展及其在数学物理等领域的应用都有着重要的意义。     

二阶椭圆与抛物型复方程的一般初非正则斜微商问题解的唯一性与先验估计

本文论述了多连通柱体区域上二阶椭圆与抛物型复方程的一般初非正则斜微商边值问题。我们先给出这个边值问题的提法,然后运用极值原理证明上述问题的解的唯一性,最后给出上述问题解的先验估计。     

非线性分数阶微分方程边值问题多重正解的存在性

 研究了非线性分数阶微分方程边值问题 ^cD^α₀₊u(t)+f(t,u(t))=0, 0cD^α₀₊为Caputo分数阶导数.通过Green函数的性质,利用不动点定理得出了奇异和非奇异微分方程边值问题多重正解的存在性的一些理论以及奇异问题的唯一解存在性理论,并给出了相应的例证.     

一类带非局部源的反应扩散方程解的整体存在与爆破

研究了一类具有非局部内吸收和带有非线性Neumann边界条件的拟线性反应扩散方程的整体解和爆破解.通过构造出适当的辅助函数,利用改进的微分不等式技巧,首先得到了整体解存在的充分条件,然后研究了解在有限时间的爆破,同时得到了解的爆破时间t~*的上界和下界估计.     

经由上下解逼近二阶微分方程解的存在性

讨论了奇异初值边界问题。通过用一列非奇异问题且利用这一列问题算子拓扑度与上下解的关系，去逼近所讨论的奇异初值边界问题，得到解的存在性结果。     

H convergence rates of solutions of GBBM equations in multi-dimensional spaces

In this paper, we study the decay rates of the generalized Benjamin–Bona–Mahony equations in n-dimensional space. By using Fourier analysis for long wave and by applying the energy method for short wave, we obtain the H^m convergence rates of the solutions when the initial data are in the bounded subset of the phase space H^m(Rⁿ)(n3). The optimal decay rates are obtained in our results and are found to be the same as the Heat equation.     

Hamilton-Jacobi-Bellman equations and dynamic programming for power-optimization of a multistage heat engine system with generalized convective heat transfer law

A multistage endoreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with generalized convective heat transfer law [q∝(ΔT) m ] is investigated in this paper.Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations,which determine the optimal fluid temperature configurations for maximum power output under the conditions of fixed initial time and fixed initial temperature of the driving fluid.Based on the universal optimization results,the analytical solution for the Newtonian heat transfer law (m=1) is also obtained.Since there are no analytical solutions for the other heat transfer laws (m≠1),the continuous HJB equations are discretized and dynamic programming algorithm is performed to obtain the complete numerical solutions of the optimization problem.The relationships among the maximum power output of the system,the process period and the fluid temperature are discussed in detail.The results obtained provide some theoretical guidelines for the optimal design and operation of practical energy conversion systems.     

一类拟线性抛物-椭圆趋化增长系统解的全局有界性

讨论了一类拟线性抛物-椭圆趋化增长系统初边值问题,利用先验估计、Lp估计的技巧,得到了该模型解的全局存在性和有界性.     

具p-Laplacian非线性奇异边值系统正解的存在性

研究了下面带有p-Laplacian算子的非线性奇异边值系统:(ф_p(u'_i))'+a_i(t)f_i(u₁,u₂)=0, 0＜t＜1,α_iф_p(u_i(0))-β_iф_p(u'_i(0))=0, γ_iф_p(u_i(1))+δ_iф_p(u'_i(1))=0,(i=1,2)正解的存在性。其中ф_p(s)为p-Laplacian 算子，即ф_p(s)=|s|^p-2s, p>1, (ф_p)^-1=ф_q,1/p+1/q=1, α_i＞0, β_i≥0, γ_i＞0, δ_i≥0, i=1,2.这里f_{i是下半连续函数(i=1,2). 通过使用锥上的不动点定理,在相当弱的条件下, 获得了这类奇异边值系统正解的存在性.}     

一类离散三点边值共振问题解的存在性

运用紧向量场方程的解集连通理论为三点边值共振问题 Δ²u(t-1)=f(t, u(t)),t∈T, u(0)=εΔu(0), u(T+1)=αu(η) 发展上下解方法, 其中f: T×R→R 连续,T为固定的正整数, T:=｛1, 2,…,T｝, ε∈［0,∞), α∈(0,∞), η∈T 均为固定的常数, 且满足 α(η+ε)=T+1+ε。     

一类Rn上奇异非线性双调和方程正整解的存在性及性质

 以Schauder-Tychonoff不动点定理为工具,建立了一类Rⁿ上奇异非线性双调和方程Δ²_u=f(|x|,u,|▽u|)u^-β(n≥3,β>0)正的径向对称整体解的存在性定理,并给出了解的有关性质.     

一维有界区域上单稳态方程多重正解的存在性

研究了一维有界区域上第一类边界条件下的单稳态反应扩散方程,对于任意给定的正整数n,构造了一类单稳态非线性项,使得问题至少存在n个正平衡解.     

具有共振的分数阶微分方程边值问题解的存在性

通过选择恰当的Banach空间及其范数,定义合适的投影算子,利用Mawhin重合度理论和分数阶微分以及分数阶积分的性质,在Riemann-Stieltjes积分边界条件下,研究非线性项中含有分数阶导数且具有共振的分数阶(n-1,1)共轭边值问题解的存在性,其中的非线性项可以是不连续的,并给出一个例子说明了主要结论。     

Singular Integral Equations with Cosecant Kernel in Solutions with Singularities of High Order

0　IntroductionIn Ref. [1 3], Lu Jian ke first discussed singular integralequations with solutions having singularities of order oneK≡a(t0)(t0) b(t0)πi∫L(t)t- t0dt ∫Lk(t0,t)(t)dt= f(t0), t0 ∈L (1)wh