Difference Schemes with Nonuniform Meshes for Nonlinear Parabolic System

1.Introduction1.Fromtheverybeginningofsixtiestothelateeighties,therearemanywerkscontributedtothestudiesoftheboundaryproblemsandinitialvalueproblemsfortheordinarydifferentialequationsbythemethodofdifferenceschemeswithnonuniform.eshesl1-4l.Butitisextremelyrareontheworksconcerningtotheanalysisoffinitedifferenceschemeswithnonuniformmeshesfortheproblemsofpartialdifferentialequations.Byusingofthedifferenceschemeswithnonuniformmeshesapprotimationfortheproblemsofpartialdifferentialequationsthereareman…     

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.     

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.     

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one- and two-dimensional test examples.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

A collocation method for solving a class of singular nonlinear two-point boundary value problems

In this paper, an iterative algorithm for solving singular nonlinear two-point boundary value problems is formulated. This method is basically a collocation method for nonlinear second-order two-point boundary value problems with singularities at either one or both of the boundary points. It is proved that the iterative algorithm converges to a smooth approximate solution of the BVP provided the boundary value problem is well posed and the algorithm is applied appropriately. Error estimates for uniform partitions are also investigated. It has been shown that, for sufficiently smooth solutions, the method produces order h⁴ approximations. Numerical examples are provided to show the effectiveness of the algorithm.     

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

0IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblemforthefailliliarequationwherefiisaboundeddomaininR"withsmoothboulldaryoff,p22isacollstantalldlp(z,u)l5of'(tl" 'forsomea20andc>0.FOrp(x,ti)=Itll"'u,theauthorsofpaper[4,7llolwereillterestedillllollllegativesolutionandhadobtainedfollowillgresults(alsosee[12]).1)If25a 2相似文献   

Convergence of Rothe's method for fully nonlinear parabolic equations

Convergence of Rothe's method for the fully nonlinear parabolic equation u_t+F(D²u, Du, u, x, t)=0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Hölder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.     

Automatic control problems for reaction-diffusion systems

A class of nonlinear reaction-diffusion systems is considered. We formulate some automatic control problems based on feedback devices located on the boundary. Two different types of devices are analyzed: relay switch and Preisach hysteresis operator. The resulting models lead to a nonlinear integrodifferential parabolic system with nonlinear and nonlocal boundary conditions. We prove global existence and uniqueness of solutions in both the cases considered.     

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。     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.     

The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems

A kind of the general finite difference schemes with intrinsic parallelism forthe boundary value problem of the quasilinear parabolic system is studied without assum-ing heuristically that the original boundary value problem has the unique smooth vectorsolution. By the method of a priori estimation of the discrete solutions of the nonlineardifference systems, and the interpolation formulas of the various norms of the discretefunctions and the fixed-point technique in finite dimensional Euclidean space, the exis-tence and uniqueness of the discrete vector solutions of the nonlinear difference systemwith intrinsic parallelism are proved. Moreover the unconditional stability of the generalfinite difference schemes with intrinsic parallelism is justified in the sense of the continu-ous dependence of the discrete vector solution of the difference schemes on the discretedata of the original problems in the discrete w_2~(2,1) norms. Finally the convergence of thediscrete vector solutions of the certain differe     

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.      

Nonlinearly coupled thermo-visco-elasticity

The d-dimensional thermo-visco-elasticity system for Kelvin–Voigt-type materials at small strains with a general nonlinear coupling is considered. Thermodynamical consistency leads to a heat capacity dependent both on temperature and on the strain. Using higher-gradient theory, namely the concept of so-called second-grade non-simple materials (or of hyper-stresses), existence of a weak solution to a system arising after an enthalpy-type transformation is proved by a suitably regularized Rothe method, fine a-priori estimates for the temperature gradient performed for the coupled system, and a subsequent limit passage.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

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.     

On convergence to steady state of solutions of parabolic equations in unbounded domains

Solutions of the first mixed boundary value problem for a class of parabolic partial differential equations are shown to converge to solutions of associated elliptic boundary value problems as time t → ∞. The method of proof involves the introduction, for the parabolic problem, of a class of generalized solutions which may behave asymptotically, at spatial infinity, like solutions of the associated elliptic equation.     

带时滞的退化非线性抛物方程的熄灭

该文研究一带时滞的退化非线性抛物方程的初边值问题。运用正则化方法和上下解技巧证明了上述问题的古典正解的局部存在性及其可延拓性。讨论了整体存在性和 有限时刻熄灭，建立了临界长度，得到了熄灭点的位置以及特殊f(u)情形下的熄灭速率估计。     

Difference Schemes of Fully Nonlinear Parabolic Systems of Second Order

The general difference schemes for the first boundary problem of the fully nonlinear parabolic systems of second order f(x, t, u, u_x, u_{xx}, u_t) = 0 are considered in the rectangular domain Q_T = {0 ≤ x ≤ l, 0 ≤ t ≤ T}, where u(x, t) and f(x, t, u, p, r, q) are two m-dimensional vector functions with m ≥ 1 for (x, t) ∈ Q_T and u, p, r, q ∈ R^m. The existence and the estimates of solutions for the finite difference system are established by the fixed point technique. The absolute and relative stability and convergence of difference schemes are justified by means of a series of a priori estimates. In the present study, the existence of unique smooth solution of the original problem is assumed. The similar results for nonlinear and quasilinear parabolic systems are also obtained.     

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.