Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.     

Existence and uniqueness of local solutions to a class of quasilinear degenerate parabolic systems

In this paper, the authors study an initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use the method of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and hence obtain the existence of solutions to a regularized system of equations. Then the existence of solutions to the system of evolution p-Laplacian equations is obtained with the use of a standard limiting process. The uniqueness of the solution is also proven.     

Initial Value Problem for a Nonlinear Evolution System with Singular Integral Differential Terms

The initial value problem for a nonlinear evolution system with singular integral differential terms is studied. Dy means of a priori estimates of the solutions and Lcray-Schauder's fixed point theorem, we demonstrate the existence and uniqueness theorems of the generalized and classical global solutions to the problem.     

Skorohod problem and multivalued stochastic evolution equations in Banach spaces

By solving a deterministic Skorohod problem in the framework of evolutional triple, we prove the existence and uniqueness of solutions to multivalued stochastic evolution equations involving maximal monotone operators. The existence and uniqueness of invariant measures associated with the solutions as Markov processes are also considered in the present paper. Moreover, we apply the results to stochastic differential equations with normal reflecting boundary conditions and with singular drift terms, as well as a class of multivalued nonlinear stochastic partial differential equations with possibly discontinuous coefficients.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

Global existence for the full von Kármán system

We consider here the full system of dynamic von Kármán equations, taking into account the in-plane acceleration terms, which is a model for the vibrations of a nonlinear elastic plate. We prove global existence and uniqueness of strong solutions for this system with various boundary conditions possibly including feedback terms which are useful for stabilization purposes.     

高维非线性四阶抛物型方程初值问题全局解

对一类高维非线性四阶抛物型方程初值问题,在初始数据适当小及非线性项适当光滑的情况下,获得了其古典解全局存在性。     

Image filtering with the use of anisotropic diffusion

A system of nonlinear parabolic equations describing the evolution of a color image is considered. The existence and uniqueness of a global solution to the mixed problem for this system is proved.     

The Existence and Behavior of Global Solutions to a Mixed Problem with Acoustic Transmission Conditions for Nonlinear Hyperbolic Equations with Nonlinear Dissipation

A mixed problem with acoustic transmission conditions for nonlinear hyperbolic equations with nonlinear dissipation is considered. The existence, uniqueness, and exponential decay of global solutions to this problem with focusing nonlinear sources are proved Additionally, the existence of global solutions and the solution blow-up in a finite time are proved for the case of defocusing nonlinear sources.     

与年龄相关的非线性时变种群发展方程解的存在与唯一性

该文讨论了与年龄相关的非线性时变种群发展方程,证明了其局部解和整体解的存在性与唯一性及稳定性. 其结果可为种群问题的实际研究提供严格的理论基础．      

ON A SEMILINEAR MRABOLIC EQUATION SYSTEM WITH NONLOCAL AND COUPLED BOUNDARY CONDITIONS

1IntroductionInthestudyofquasi-statethermoelasticity,Deng[1-2]derivedamathemati-calmodelwhichinvolvesalinearparabolicequationwithanonlocalboundarycondition.Thismodelhasbeenextendedtomoregeneralsemilinearparabo-licequationsinhigh-dimensiondomainsbyFriedman[5]andKawohlI6],andmorerecentlybyDeng[3],Yin[13],Paol8-lo]andWang[11],andvariouscomparison,estimateandstabilityresultshavebeenobtained.InhtispaperweextendtheproblemofPao[9]tothefollowingproblemwithmoregeneralcoupledboundaryconditions(PE):w…     

一类带非局部源项的p-Laplace方程解的整体存在与爆破

本文研究一类在Neumann边值条件下带局部源项的p-Laplace方程解的整体存在和爆破性.利用微分不等式技巧,通过构造辅助函数的方法,获得了方程的解整体存在和解在有限时间爆破的充分条件,以及爆破时间的上下界估计,推广了相关文献结论.     

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.     

强阻尼波方程解的整体存在性和一致衰减性

这篇文章研究一类带非线性源项和非线性边界阻尼项的强阻尼波 方程强解和弱解的整体存在性和唯一性,进而也讨论解的一致衰减.     

关于一类非线性发展方程整体解的存在性问题

本文研究一类模拟非线性弹性杆的纵振动的非线性发展方程的初边值问题，证明了其整体解的存在性、唯一性、光滑性及在一定条件下，整体解的不存在性。     

Global solutions to thermomechanical equations with nonconvex Landau-Ginzberg free energy

The global existence and uniqueness of classical solutions is proved for the system of partial differential equations describing the nonlinear thermomechanical processes with nonconvex Landau-Ginzberg free energy.This work was partially supported by National Science Foundation of China.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay, in which the initial value belongs to the abstract space ℬ. We establish the existence and uniqueness of mild solutions for SNSEEIs under global and local Carathéodory conditions by means of the successive approximation. An application to the stochastic nonlinear wave equations with infinite delay is given to illustrate the theory.     

Semibounded Nonlinear Evolution Equations with Application to the KdV Equations

Existence of solutions for semibounded nonlinear evolution equations is established. This gives more accurate estimate of solutions and conditions of existence axe more easily validated. Our results are successfully applied to prove existence and uniqueness of solutions for some KdV type equations.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.