Existence of solutions for a nonlocal epitaxial thin film growing equation

The existence of weak solutions is studied to the initial boundary problem of a nonlocal epitaxial thin film growing equation modeling epitaxial thin film growth. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Existence and Uniqueness of Weak Solutions to the $p$-Biharmonic Parabolic Equation

We consider an initial-boundary value problem for a $p$-biharmonic parabolic equation. Under some assumptions on the initial value, we construct approximate solutions by the discrete-time method. By means of uniform estimates on solutions of the time-difference equations, we establish the existence of weak solutions, and also discuss the uniqueness.     

具变指数退化四阶抛物方程弱解的存在性

考虑一类具变指数退化四阶抛物方程的初边值问题.在一些初值的假定下,基于时间离散化方法构造逼近解,通过对逼近解的一致性估计,证明了弱解的存在性.     

一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.     

A sixth order degenerate equation with the higher order <Emphasis Type="Italic">p</Emphasis>-laplacian operator

We consider a initial-boundary value problem for a sixth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the time-discrete method. The uniqueness, asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

A class of degenerate parabolic equations with a concentrated nonlinear source

In this paper, we consider the Cauchy problem for a class of degenerate parabolic equations with a concentrated nonlinear source. We obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions.     

Asymptotic Behaviour and Global Existence of Solutions for Some Classes of Nonlinear Parabolic Equations

The paper deals with the asymptotic behaviour and global existence of solutions for some classes of nonlinear parabolic equations in regard to the monotone properties of the nonlinear term. The asymptotic behaviour of the solutions of initial-boundary value problem for nonlinear parabolic equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions. Existence of extremal solutions of semilinear elliptic and parabolic equations is investigated via monotone iterative methods. The extremal solutions are obtained via monotone iterates.     

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

Variational inequalities for a system of elliptic–parabolic equations

We create a general framework for mathematical study of variational inequalities for a system of elliptic–parabolic equations. In this paper, we establish a solvability theorem concerning the existence of solutions for the vector-valued elliptic–parabolic variational inequality with time-dependent constraint. Moreover, we give some applications of the system, for example, time-dependent boundary obstacle problem and time-dependent interior obstacle problem.     

Theoretical analysis of a model of the thermal boundary layer with drag

For solutions of a system of degenerate quasilinear parabolic equations we prove some interior Schauder estimates and use them to establish an existence theorem for solutions of the problem of extending the thermal boundary layer of a compressible fluid in a magnetic field.     

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     

Existence of positive radial solutions for a nonlinear elliptic problem in annulus domains

In this paper, we study a nonlinear elliptic problem in an annulus domain. For which, the nonexistence of nontrivial solutions has been obtained by some authors in star‐shaped domain. Although we note that it may admit non‐trivial solutions if the domain is non‐star shaped. With the use of a variational method, we establish the existence of positive radial solutions in an annulus domain. On the basis of this, we also extend this result to the periodic problem of the corresponding degenerate parabolic problem. Copyright © 2012 John Wiley & Sons, Ltd.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

带有可测系数的二阶非线性抛物型方程组的初—混合边值问题

在多连通区域上研究带有可测系数的二阶非线性抛物型方程组的初－混合边值问题，首先我们将其化为复形式的方程组，并给出在一定条件下的上述初－这值问题解的先验估计，然后利用解的估计和列紧性原理，证明了这种初－边值问题解的存在性。在论证过程中，我们始终用复分析方法讨论文中所提出的问题，没有看到国外有人使用这种方法处理此类问题。     

A parabolic problem arising in Financial Mathematics

We study a parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence and uniqueness of solutions in a general domain using the method of upper and lower solutions and a diagonal argument.     

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.     

Insensitizing controls for a class of quasilinear parabolic equations

This paper is addressed to showing the existence of insensitizing controls for a class of quasilinear parabolic equations with homogeneous Dirichlet boundary conditions. As usual, this insensitizing problem is reduced to a nonstandard null controllability problem of some nonlinear cascade system governed by a quasilinear parabolic equation and a linear parabolic equation. Nevertheless, in order to solve the later quasilinear controllability problem by the fixed point technique, we need to establish the null controllability of the linearized cascade parabolic system in the framework of classical solutions. The key point is to find the desired control function in a Hölder space for given data with certain regularities.