具有间断非线性项的拟线性抛物型方程(英)

本文研究具有间断非线性项的拟线性抛物型方程，利用Clarke广义梯度和伪单调算子理论证明了解的存在性．     

具有间断非线性项的拟线性抛物型方程

本文研究具有间断非线性项的拟线性抛物型方程，利用Ｃｌａｒｋｅ广义梯度和伪单调算子理论证明了解的存在性。     

带梯度项的双重退缩抛物方程正解的Blow-up

研究了RN中一般区域上的一族带非线性梯度项的双重退缩抛物方程解的Blow-up性质.通过构造适当的辅助函数,利用特征函数法和不等式技巧,给出了其齐次Dirichlet边值问题的正解产生Blow-up的充分条件:利用能量方法,证明了其Cauchy问题非平凡整体解的不存在性.方法也适用于研究其它带非线性源的退缩非线性抛物方程解的Blow-up问题.     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

一类非线性伪抛物型方程Cauchy问题的适定性

通过引入算子I-Δ的Bessel势将伪抛物型方程化成抽象的抛物型方程,然后利用算子半群理论讨论了一类非线性伪抛物型方程Cauchy问题的适定性问题.     

一类非线性脉冲中立抛物型分布参数系统的振动条件

研究一类带中立项及高阶Laplace算子的非线性脉冲抛物型分布参数系统在第一类边值条件下的振动性问题,利用处理中立项及高阶Laplace算子的技巧和积分平均方法,建立了该类系统所有解振动的若干新的充分性条件.所得结论充分表明系统振动是由脉冲量和时滞量引起的.     

具混合边界的双曲型微分方程非平凡解的存在性[英文]

本文将具混合边界的一类双曲型微分方程分解为两个线性算子和三个非线性算子.证明了这些算子具有单调性质,由此得到一类算子方程存在解的结论,进而证明具混合边界的双曲型非线性微分方程存在唯一非退化解的结论.此文是对含有p-Laplacian算子的非线性椭圆和非线性抛物方程相关研究工作的推广,并采用了一些新的证明技巧.     

粗糙抛物次线性算子和交换子的抛物广义局部Morrey空间估计（英文）

引入了抛物广义局部Morrey空间,并得到了其上一大类抛物粗糙算子的有界性.还创建了其在抛物广义局部Morrey空间上交换子的抛物局部Campanato空间估计·带粗糙核的抛物次线性算子及其交换子的对应结果可作为特例而得到,作为应用,得到了抛物广义局部Morrev空间上一些解析半群抛物分数次幂的有界特征.     

带极大单调图的半线性抛物型方程的零能控性(英文)

本文讨论非线性项带一个极大单调图的半线性抛物型方程系统的零能控性.文中利用Kakutani不动点定理和线性抛物型方程的能控性得到该系统是零能控的,如果控制作用在内部区域上.由此,还得到该系统是零能控的,如果控制施加在一部分边界上.     

微伸缩的热弹性力学方程组柯西问题解的奇性传播

用频率分析对角化的方法,研究了一维线性微伸缩的热弹性力学方程组柯西问题解的奇性传播规律.首先从微局部观点出发,利用拟微分算子将双曲抛物的耦合方程组弱解耦.然后利用经典的双曲抛物方程理论和穿梭法,证明了柯西问题解的奇性传播具有有限传播速度、解的奇性沿双曲算子的零次特征带进行传播.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Numerical implementation of iteration processes applied in the study of nonlinear boundary value problems of the theory of elasticity and filtration

Nonlinear elastic problems for hardening media are solved by applying the universal iteration process proposed by A.I. Koshelev in his works on the regularity of solutions to quasilinear elliptic and parabolic systems. This requires numerically solving a linear elliptic system at each step of the iteration procedure. The method is numerically implemented in the MATLAB environment by using its PDE Toolbox. A modification of the finite-element procedure is suggested in order to solve a linear algebraic system at each iteration step. The computer model is tested on simple examples. The same nonlinear problems are also solved by the method of elastic solutions, which consists in replacing the Laplace operator in the universal iteration process by the Lamé operator of linear elasticity. As is known, this iteration process converges to a weak solution of the nonlinear problem, provided that the displacements are fixed on the boundary. The method is tested on examples with stresses on the boundary. The third part of the paper is devoted to the nonlinear filtration problem. General properties of the iteration process for nonlinear parabolic systems have been studied by A.I. Koshelev and V.M. Chistyakov. The numerical implementation is based on slightly modified PDE Toolbox procedures. The program is tested on simple examples.     

Suboptimality Estimates for the Semi-Discretized LQR Problem for Parabolic PDEs

The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite-dimensional Hilbert space equivalent of the finite-dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite-dimensional case become operator equations in the infinite-dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite-dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

Separation of linear components in nonlinear boundary heat control

By means of an additional substitution a parabolic control problem with some nonlinear boundary condition will be decoupled into some control problem with linear parabolic state equations and an appropriate nonlinear mapping. This separation allows the use of efficient techniques e.g. Fourier methods, to determine the solution of linear parabolic state equations. Essential properties of the mapping used in the transformation are studied. Further, the application of piecewise constant discretizations of the controls in connection with the proposed splitting is discussed.     

Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type

In this paper, the qualitative properties of the solutions of the Cauchy problem for degenerate parabolic equations with a nonlinear operator of Baouendi-Grushin type are studied. Sharp local and global (with respect to the spatial and temporal variables) estimates of the solution are obtained. The property of the finiteness of the support of the solution is established.     

Linear parabolic problem: High-frequency asymptotics in the critical case

A second-order linear parabolic problem with high-frequency terms is considered. The elliptic operator of the corresponding limiting (averaged) problem is assumed to be degenerate. A complete formal asymptotic expansion of a time-periodic solution of the perturbed problem is constructed.