四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

一类奇异摄动扩散方程的正则半群的吸引子的上半连续性

该文对一类奇异摄动扩散方程u\-t-εΔu\-t-Δu=f(u)的正则半群的渐近行为进行了研究。 得到了其全局吸引子的存在性及全局吸引子在ε=0的上半连续性.     

不可微泛函拟极小问题解的正则性

本文对P＞1的情形给出不可微泛函I(u,G)= ∫FG（x,u, ↓△）dx的(1 cR^2u）一极小问题解梯度的Holder连续性证明。     

一类椭圆方程很弱解关于区域的连续性

本文讨论了二阶椭圆型方程-△u=f(x,u),x∈Ω的Dirichlet问题u | Ω=0的很弱解u∈W ,r(Ω)(1＜r＜2)关于区域Ω的连续性及很弱边值问题的很弱解的唯一性.     

有界凸域上拟共形映射的Holder连续性

该文利用曲线族的模,得到了n维空间中的有界凸域 D 到单位球 Bⁿ上的 K -拟共形映射f的全局Holder连续性, 且Holder指数α=K^1/(1-n)是最佳的.     

路和圈多重联图的邻点可区别E-全染色

设G（V,E）是一个简单图,k是一个正整数,f是一个V（G）∪E（G）到{1,2,...,k}的映射.如果u,v∈E（G）,则f（u）=f（v）,f（u）=f（uv）,f（v）=f（uv）,C（u）=C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.称f是图G的邻点可区别E-全染色,称最小的数k为图G的邻点可区别E-全色数.讨论了路和圈的多重联图的邻点可区别E-全色数。     

含不连续项的常微分积分方程的解

本文研究微分积分方程 u'=g(t,u)+integral from 0 to 1(k(t,s)f(s,u(s))ds),u(0)=x_0最小解、最大解的存在性.本文的特点是关于方程中函数g(t,x),f(t,x)没作任何连续性假定.     

一类拟线性退化抛物方程的Cauchy问题

对方程α（xx）u＋uαyu-αtu=f（·,u）,（x,y,t）∈R^2×（0,T）的Cauchy问题,给出了一个整体BV-弱解的定义,证明了该弱解的连续性.     

数形结合解复合函数的零点个数的常见解法

本文通过利用函数图像的方法研究复合函数y=g（f（x））的零点问题,即复合函数方程g（f（x））=0的根,令u=f（x）（内层方程）,这样g（f（x））=0就转化成g（u）=0.当外层方程g（u）=0容易求解时,可以先解方程g（u）=0,再解内层方程u=f（x）,这样方程的总个数即为复合函数y=g（f（x））的零点个数.     

有限图上高阶Yamabe型方程的非平凡解

该文研究了以下高阶Yamabe型方程Lm,pu - g|u|p-2 u = λ f|u|α-2 u在有限图上的非平凡正解的存在性,其中Lm,p是一个2m阶差分算子,它是一种p次(-Δ)m算子更一般化,α≥p≥2,g＞0和f ＞0是定义在G的所有顶点上的实函数,m≥1是一个整数.     

HOLDER ESTIMATES FOR SOLUTIONS OF UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS

In this paper the author discusses the quasilinear parabolic equation $$\[\frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial {x_i}}}[{a_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}] + {b_i}(x,t,u)\frac{{\partial u}}{{\partial {x_i}}} + c(x,t,u)\]$$ Which is uniformly degenerate at $\[u = 0\]$. Let $\[u(x,t)\]$ be a classical solution of the equation satisfying $\[0 < u(x,t) \le M\]$. Under some assumptions the author establishes the interior estimations of Holder coefficient of the solution for the equation and the global estimations for Cauchy problems and the first boundary value problems, where Holder ooeffioients and exponents are independent of the lower positive bound of $\[u(x,t)\]$.     

THE NAGUMO EQUATION ON SELF－SIMILAR FRACTAL SETS

The Nagumo equationut ut=△u+bu(u-a)(1-u),t＞0 is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the“pathological” property of the fractal. However, it is shown that a global Hoelder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl‘s formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.     

本刊英文版Vol.37(2021),No.11论文摘要（英文）

<正>p-Laplacian Equations on Locally Finite Graphs Xiao Li HAN Meng Qiu SHAO Abstract This paper is mainly concerned with the following nonlinear p-Laplacian equation-Δ_pu(x)+(λa(x)+1)|u|~(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E) with more general nonlinear term,whereΔ_p is the discrete p-Laplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution u_λvia the method of Nehari manifold,for anyλ> 1.In addition,asλ→+∞,we prove that the solution u_λconverge to a solution of the following Dirichlet problem     

一类色散方程的L~p-Besov估计

设u(x,t)=(SΩf)(x,t)是一般色散初值问题(?)tu-iΩ(D)u=0,u(x,0)=f(x),(x,t)∈Rn×R的解,SΩ*f,SΩ**f是它的局部和整体极大算子.本文给出它们范数的若干估计.     

EXISTENCE AND UNIQUENESS OFWEAK SOLUTIONS OF UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS

In this paper we deal with the quasilinear parabolic equation u/t=/x_i[a_(ij)(x, t, u))u/x_j]+b_i(x, t, u)u/x_i+c(x, t, u) which is uniformly degenerate at u=O. Under some assumptions we prove existence anduniqueness of nonnegative weak solutions to the Cauchy problem and the first boundary valueproblem for this equation. Furthermore, the weak solutions are globally Holder continuous.     

An Inverse Problem for the Heat Equation II

Completeness of the set of products of the derivatives of the solutions to the equation ( av ')' m u v = 0, v (0, u ) = 0 is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $ \dot u = (a(x)u')' $ , u ( x , 0) = 0, u (0, t ) = 0, u (1, t ) = f ( t ) known for all t > 0, from the heat flux a (1) u '(1, t ) = g ( t ). Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point x = 0 in our case) does not have a unique solution, in general.     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）在ｕ０有界可测，ｆ连续且ｆ＞０条件下，其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）的熵解，即论证了“沾性消失法”解此方程的正确性，１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

一类非散度型椭圆方程的正则性

本文得到了一类带奇异低阶项椭圆万程的非负解的Harnack不等式、方程的形式为L0u+biuxi=0,其中L0为一具Holder连续系数的非散度型椭圆算子,|b|2属于Kato类.