几类二次三项式偏微分方程的整函数解

文章主要运用值分布和偏微分方程特征方程方法研究了几类一阶、二阶以及混合型偏微分方程的整函数解,获得了涉及几类二次三项式偏微分方程有限级超越整函数解的存在性及其形式,推广了先前的结果,同时举例说明所得方程解的形式是准确的.     

一类高阶线性微分方程解的增长性

研究整函数系数高阶线性微分方程f~((k))+A_(k-1)f~((k-1))+…+A_0f=0解的增长性.利用亚纯函数的Nevanlina值分布理论,得到当系数A_s(s≠0)为满足杨不等式极端情况的整函数,A_0满足一定条件时,上述方程的每个非零解均为无穷级,并给出解的超级估计.     

费马q-差分微分方程整函数解的增长性研究

本文研究了费马q-差分微分方程的整函数解的相关问题.利用经典和差分的Nevanlinna理论和函数方程理论的研究方法,获得了q-差分微分方程整函数解增长性的几个结果.     

复线性微分方程解的导数的Julia集的径向分布

本文主要研究一类复线性微分方程的整函数解的导数的Julia集的径向分布.在适当条件下,本文证明这类复微分方程的整函数解及其导数的Julia集具有相似的径向分布,并找到了它们的下界,从而改进了最近的一些相关结果.     

二阶复域微分方程解的不动点与超级

文中首次研究了４种类型的整函数系数的二阶线性微分方程的解的不动点及超级问题,得到：复域微分方程解的不动点性质,由于受到微分方程的制约,与一般超越整函数的不动点性质相比,是十分有趣的．     

关于二阶复微分方程解的幂和解的微分多项式的不动点

本文研究了以整函数为系数的二阶线性微分方程解的幂和解生成的微分多项式的不动点问题,得到:二阶线性微分方程解生成的微分多项式的不动点性质,由于受到微分方程的限制,与一般亚纯函数的不动点性质相比是十分有趣的.事实上,它们与解的增长性密切相关.     

关于高阶整函数系数微分方程解的增长性

研究了两类高阶线性整函数系数微分方程解的增长性，得到了其超级的精确估计，文中的定理改进了G.Gundersen和陈宗煊的有关结果。     

复非线性代数微分方程解的增长级(英文)

本文研究了一类非线性微分方程的解的增长级的问题.利用亚纯函数的Nevanlinna值分布理论和Wiman-Valiron整函数理论的方法,获得了比以往文献更为精确,更为一般的结论,推广了Gol’dberg,Barsegian,Hayman以及Korhonen等作者的一些结果.     

一类高阶复微分方程解的增长性

本文研究高阶线性微分方程f~((k))+A_(k-1)f~((k-1))+···+A1f′+A0f=0解的增长性,其中Aj(j=0,···,k-1)为整函数.当存在某个系数A_s是方程ω′′+P(z)ω=0的一个非零解时,我们得到上述方程具有无穷级解的判定条件,并对解的超级进行了估计.这里的P(z)为非零多项式,当P(z)为特定形式的多项式时,A_s可取为Airy函数,Weber-Hermite函数或指数函数.     

一类二阶整函数非齐次线性微分方程的复振荡

研究具有整函数函数系数的二阶非齐次线性微分方程:f″+A(z)e~(az)f′+B(z)e~(P(z))f=F(z)解的复振荡,其中P(z)为非常数多项式且deg(P)=n,A(z),B(x),F(z)均为整函数且max{ρ(A),ρ(B)}n.我们将看到方程的任一非零解具有无穷增长级.     

Entire solutions of delay differential equations of Malmquist type

The celebrated Malmquist theorem states that a differential equation, which admits a transcendental meromorphic solution, reduces into a Riccati differential equation. Motivated by the integrability of difference equations, this paper investigates the delay differential equations of form $w(z+1)-w(z-1)+a(z)\frac{w''(z)}{w(z)}=R(z, w(z))(*),$ where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solution with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc. Amer. Math. Soc. 145, no.6 (2017)], we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The value distribution and forms of transcendental entire solutions for the reduced delay differential equations are studied. The existence of finite iterated order entire solutions of the Kac-van Moerbeke delay differential equation is also detected.     

Hamiltonian constants for several new entire solutions

Using the Hamiltonian identities and the corresponding Hamiltonian constants for entire solutions of elliptic partial differential equations, we investigate several new entire solutions whose existence were shown recently, and show interesting properties of the solutions such as formulas for contact angles at infinity of concentration curves.      

Remarks on Third-Order ODEs Relevant to the Kuramoto-Sivashinsky Equation

We are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.     

Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.     

Entire solutions of integrodifference equations

This paper is concerned with the existence of nontrivial entire solutions of integrodifference equations. By constructing proper auxiliary integrodifference equations and functions, we present the existence of nontrivial entire solutions even if the birth function is not monotone, which are different from the well-studied travelling wave solutions. The convergence of entire solutions is also given.     

A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials

We have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations. Specifically, we deal with entire solutions of the equations that are represented in convergent series of Bessel polynomials, and determine orders and types of the solutions, in terms of their Taylor coefficients, by establishing an analogue of Lindelöf-Pringsheim theorem as well as Wiman-Valiron type theory for such functions. Finally, by using value distribution theory of holomorphic functions, we are able to exhibit some uniqueness theorems of the entire (or meromorphic) solutions.     

Function solutions to certain Diophantine equations

We investigate whether certain Diophantine equations have or have not solutions in entire or meromorphic functions defined on a non-Archimedean algebraically closed field of characteristic zero. We prove that there are no non-constant meromorphic functions solving the Erdös–Selfridge equation except when the corresponding curve is a conic. We also show that there are infinitely many non-constant entire solutions to the Markoff–Hurwitz equation.     

一类无穷下级整函数的Julia集的径向分布

主要研究方程f"(z)+A(z)f'(z)+B(z)f(z)=0(A(z)),B(z)为整函数)的解、解的多项式或微分多项式这些具有无穷下级的整函数的Julia集的径向分布问题.     

薛定谔方程及薛定谔-麦克斯韦方程的多解

不作周期性和对称性的假设,也没有Ambrosetti-Rabinowitz增长控制条件,我们得到了一类超线性薛定谔方程在全空间中无穷多解的存在性结果.同时,得到了一类超线性薛定谔-麦克斯韦方程无穷多解的存在性结果.     

Large solutions of coupled sublinear/superlinear elliptic equations

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u ^α + q(x)v ^β on bounded domains in R ⁿ , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.