INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS

This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.

Hlder Continuity of the Gradient of Solutions of Nonlinear Degenerate Parabolic Systems

§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if p>2 or singular if 1<p<2. A vector function u=(u1, u2, …, um) defined in ΩT=Ω×[0, T] is called a solution of the system (1.1) if

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.

Backstepping-Based Distributed Abnormality Detection for Nolinear Parabolic Distributed Prameter Systems

In this paper, we proposed a model-based abnormality detection scheme for a class of nonlinear parabolic distributed parameter systems (DPSs). The proposed methodology consists of the design of an observer and an abnormality detection filter (ADF) based on the backstepping technique and a limited number of in-domain measurements plus one boundary measurement. By taking the difference between the measured and estimated outputs from observer, a residual signal is generated for fault detection. For the detection purpose, the residual is evaluated in a lumped manner and we propose an explicit expression for the time-varying threshold. The convergence properties of the PDE observer and the residual are analyzed by Lyapunov stability theory. Eventually, the proposed abnormality detection scheme is demonstrated on a nonlinear DPS.

STABILITY OF DIFFERENCE SCHEMES FOR NONLINEAR PARABOLIC SYSTEMS WITH THREE DIMENSIONS

In this paper the existence of discrete vector solutions with bounded second order quotientsfor the difference systems of nonlinear parabolic system is established by the fixed point technique,and then the absolute and relative stability for the general difference schemes is justified in thesense of continuous dependence of the discrete vector solution of the difference schemes on thediscrete data of the original problems.

GLOBAL EXISTENCE AND NONEXISTENCE FOR A DOUBLY DEGENERATE PARABOLIC SYSTEM COUPLED VIA NONLINEAR BOUNDARY FLUX

This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel＇dovich-Kompaneetz-Barenblatt profile

ASYMPTOTIC BEHMIOUR IN A REACTION-DIFFUSION SYSTEM

This paper proves the asymptotic behaviour for a class of reaction-diffusionsystem in bacteriology by using duality technique, semigroup theorem, Lp--estimates andupper and lower solutions method.

Regional Boundary Observability with Constraints of the Gradient

The aim of this paper is to explore the concept of observability with constraints of the gradient for distributed parabolic system evolving in spatial domain Ω, and which the state gradient is to be observed only on a part of the boundary of the system evolution domain. It consists in the reconstruction of the initial state gradient which must be between two prescribed functions in a subregion Γ of Ω. Two necessary conditions are given. The first is formulated in terms of the subdifferential associated with a minimized functional, and the second uses the Lagrangian multiplier method. Nu-merical illustrations are given to show the efficiency of the second approach and lead to open questions.

Adaptive neural networks control for uncertain parabolic distributed parameter systems with nonlinear periodic time-varying parameter

This paper studies the problem of adaptive neural networks control(ANNC) for uncertain parabolic distributed parameter systems(DPSs) with nonlinear periodic time-varying parameter(NPTVP). Firstly, the uncertain nonlinear dynamic and unknown periodic TVP are represented by using neural networks(NNs) and Fourier series expansion(FSE), respectively. Secondly, based on the ANNC and reparameterization approaches, two control algorithms are designed to make the uncertain parabolic DPSs with NPTVP asymptotically stable. The sufficient conditions of the asymptotically stable for the resulting closed-loop systems are also derived. Finally, a simulation is carried out to verify the effectiveness of the two control algorithms designed in this work.

OSCILLATIONS OF SYSTEMS OF PARABOLIC EQUATIONS WITH CONTINUOUS DISTRIBUTED ARGUMENTS

Sufficient conditions are established for the oscillations of systems of parabolic equations with continuous distributed deviating arguments of the form where Ω is a bounded domain in Rn with piecewise smooth boundary эΩ, △is the Laplacian in Euclidean n-space Rn, and the integral in (1) is a Stieltjes integral.

Existence of Periodic Solutions of Nonlinear Systems with Nonlinear Boundary Conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions.By constructing the Poincare operator,we obtain the existence of Wp2β-periodic weak solutions under some reasonable assumptions.

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.

Recent Progress in the Lp Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper,we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients.We begin with an approach given by N.V.Krylov to parabolic equations in the whole space with VMOx coefficients.We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions,weighted Lp estimates with Muckenhoupt(Ap)weights,non-local elliptic and parabolic equations,as well as fully nonlinear elliptic and parabolic equations.