Finite-dimensional controller design for semilinear parabolic systems

This paper considers the stabilization of steady-state solutions of a semilinear parabolic system using finite-dimensional feedback controllers with support in an arbitrary open subset and which are active in one equation only. It is shown that such a controller, with dimension given by the largest algebraic multiplicity of the unstable eigenvalues of the linearized system, exponentially stabilizes the steady-state solution. An optimal design methodology for these types of controllers, which is based on the finite element approximation of the semilinear parabolic system, is introduced and illustrated by numerical simulation examples.     

Schauder Estimates for Elliptic and Parabolic Equations

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.     

Sobolev Regularity for t > 0 in Quasilinear Parabolic Equations

We establish a regularity property for the solutions to the quasilinear parabolicinitial-boundary value problem (1.4) below, showing that for t > 0 they belong to the same space to which the solutions of the second order hyperbolic problem (1.5), which is a singular perturbation of (1.4), belong. This result provides another illustration of the asymptotically parabolic nature ofproblem (1.5), and would be needed to establish the diffusion phenomenon for quasilinear dissipative wave equations in Sobolev spaces.     

Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

Carleman Estimates for Parabolic Equations with Nonhomogeneous Boundary Conditions

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.     

A Remark on the Sobolev Regularity of Classical Solutions to Uniformly Parabolic Equations

We prove that C^2+α,1+α/2 (Q?) solutions of problem (1.6) below are in a subspace H_c^m+2(Q) of H^{m+2,(m+2)/2(Q)} for all m ∈ ?, if f and the coefficients are in H_c^m(Q)∪C^α,α/2 (Q?). We apply this result to obtain global existence of Sobolev solutions to the quasilinear problem (1.26) below.     

半线性抛物型积分—微分方程的谱方法

本文在讨论带有半线性记忆项抛物型方程半离散格式的基础上,构造了空间方向谱离散,时间方向后欧拉方法的全离散格式,并给出了误差估计。对于记忆项的数值积分,采用了梯形公式与矩形公式结合的方法,并估计了数值积分的影响。     

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．     

半线性抛物型泛函微分方程的周期解

本文应用上下解的概念及schauder不动点定理,在较广泛的条件下,解决了半线性抛物型泛函数微分方程周期解的存在性问题.所得结果推广了文献[3]的定理1.2.     

Some Mixed Problems for Semilinear Parabolic type Equation

In this paper, some mixed problem with third type boundary value for a semilinear parabolic equation is investigated. Here the solvability theorems for considered problem and the uniqueness theorem for a model case of the problem are showed.     

具有Neumann边界的抛物型方程组的反馈能稳性

该文研究了一类半线性抛物型方程组平衡解的有限维反馈能稳性, 控制器由对应于平衡解的线性化系统的LQ问题所导出. 该文发现在受控系统中如何施加控制是与平衡解有关的, 对于不同的平衡解, 在受控系统中应施加不同的控制.     

Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction--diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman type.     

一类微分项中含Hysteresis的抛物型方程

本文通过引入导出算子分离Ｈｙｓｔｅｒｅｓｉｓ与微分,证明了一个微分项中含Ｈｙｓｔｅｒｅｓｉｓ的抛物型方程古典解的存在唯一性。     

Necessary and Sufficient Conditions for Oscillation of Delay Parabolic Differential Equations

Necessary and sufficient conditions for oscillation of delay parabolic differential equations are obtained.AMS Subject Classification (2000): 35B05, 35R10.This work is supported by the Natural Science Foundation of Shandong Province, China (Y2001A03).     

非线性脉冲时滞抛物型偏微分方程的强迫振动性

考虑一类具强迫项的非线性脉冲时滞抛物型偏微分方程，利用脉冲时滞微分不等式，获得了该类方程的解强迫振动的若干充分判据。     

一类半线性椭圆方程的正径向解

本文以关于非线性全连续算子的锥不动点定理为工具，研究半线性椭圆边值问题上Δu λa(|x|)u f(|x|,u)=0(x∈Ω)，u|=0及Δu λf(|x|,u)=0(x∈Ω)，u|=0．在不假定f单调的情况下，本文得出了上述问题存在正径向解的若干充分条件．     

中立型抛物方程解的振动性与渐近性

本文研究了,具有分布滞量的非线性中立型抛物方程（E）解的振动性与渐近性,我们的结果推广了Bainov和Petrov[1]对方程的一些结果。     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.