Space-time Estimates for Parabolic Type Operator and Application to Nonlinear Parabolic Equations

In this present paper we establish space-time estimates of solutions for linear parabolic type equations based on classical multipliers theory or operator semigroup theory. According to space-time estimates we first construct suitable work space L^q(0, T; L^P), moreover we study the Cauchy problem and initial boundary value problem for semilinear parabolic equation in L^q(0, T; L^P) type space.     

一类非线性抛物方程弱解的L~2 衰减(英文)

本文研究了非线性抛物方程弱解的L2 衰减 ,证明了若u是非线性抛物方程的一个弱解 ,则u满足‖u‖L2 ≤C( 1 t) -n/ 4,其中C是常数且仅依赖于初始值u0 的范数 .     

Fully Nonlinear Parabolic Equations and the Dini Condition

Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolic equations under the Dini condition, which improve and generalize a result due to Kovats, are obtained by the use of the approximation lemma. Received November 21, 2000, Accepted April 27, 2001     

Generalized Solutions of Nonlinear Parabolic Equations and Diffusion Processes

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.      

Regularity of Solutions to Nonlinear Parabolic Equations with a Lower-Order Term

In this paper, we give some regularity results of solutions to nonlinear parabolic equations with a lower-order term and inregular data.     

Existence of Solutions To Weak Parabolic Equations For Measures

Let A = (a^ij) be a Borel mapping on [0, 1] x R^d with valuesin the space of non-negative operators on R^d and let b = (bⁱ)be a Borel mapping on [0, 1] x R^d with values in R^d. Let Under broad assumptions on A and b, we construct a family µ= (µ_t)_t [0, 1] of probability measures µ_t on R^dwhich solvesthe Cauchy problem L* µ = 0 with initial conditionµ₀ = , where \nu is a probability measure on R^d, in thefollowing weak sense: and Such an equation is satisfied by transition probabilities ofa diffusion process associated with A and b provided such aprocess exists. However, we do not assume the existence of aprocess and allow quite singular coefficients, in particular,b may be locally unbounded or A may be degenerate. An infinite-dimensionalanalogue is discussed as well. Main methods are L^p-analysiswith respect to suitably chosen measures and reduction to theelliptic case (studied previously) by piecewise constant approximationsin time. 2000 Mathematics Subject Classification 35K10, 35K12,60J35, 60J60, 47D07.     

非线性抛物积分微分方程的各向异性有限元高精度分析

本文讨论非线性抛物积分微分方程的各向异性有限元方法.在不引入真解的H1-Volterra投影的情况下得到了半离散格式下的整体超收敛.     

Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed. If the implicit scheme is one of the backward differentiation formulae (BDF) of order up to six, then the corresponding implicit–explicit method of the same order is stable provided the stability constant is less than a specific scheme-dependent constant. Based on BDF, implicit methods are constructed such that the corresponding implicit–explicit scheme of the same order exhibits improved stability properties.     

A Note on Three Classes of Nonlinear Parabolic Equations with Measurable Coefficients

In this note a note and simple technique is presented to replace the complicated one in [1] to obtain Hölder continuity of the weak solutions for a class of nonlinear parabolic equations with measurable coefficients, whose prototype is the singular p-Laplacian. This new approach is also applied to two other classes of nonlinear parabolic equations with measurable coefficients, whose weak solutions exhibit the similar property to those of equations mentioned above.     

Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rⁿ of the quasilinear singular parabolic equations: u_t = div(|∇u|^m − 1∇u) + t^s|x|^σu^p, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that p_c ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ p_c then every non-trivial solution blows up in finite time, but for p > p_c, a small positive global solution exists.     

Forced Oscillation for Certain Nonlinear Delay Parabolic Equations

In this paper we investigate the forced oscillation of the solution of a class of nonlinear parabolic equations with continuous distributed deviating arguments.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

In this paper we study the homogenization of degenerate quasilinear parabolic equations: where a(t, y, a, λ) is periodic in (t, y).     

A Free Boundary Problem Governed by Nonlinear Degenerate Equations of Parabolic Type

We consider a free boundary problem connected with non-Newtonian fluid motion, i.e. the flow of power law fluids with the yield stress. We obtain the solution of the relevant approximation problem by means of a parabolic quasi-variational inequality, and then obtain the weak solution of the original problem after a passage to the limit. Finally, we study the regularity of the weak solution.     

非线性抛物方程的质量集中非协调元分析

将一个低阶Crouzeix-Raviart型非协调三角形元应用到一类非线性抛物方程,并建立了质量集中的半离散和向后Euler全离散逼近格式,在一般各向异性网格上利用插值算子导出了L²-模的最优误差估计.     

区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

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.     

具非线性扩散项的脉冲抛物方程的强迫振动性

研究一类具非线性扩散项的脉冲抛物型方程解的强迫振动性问题,借助平均技巧和脉冲微分不等式,建立了该类方程在Dirichlet边值条件下所有解强迫振动的若干新的充分判据.结论充分表明振动是由脉冲效应引起的.     

非线性中立型抛物微分方程解的振动性

This paper deals with the oscillatory properties of a class of nonlinear neutral parabolic partial differential equations with several delays. Sufficient criteria for the equation to be oscillatory are obtained by making use of some results of first-order functional differential inequalities. These results fully reveal the essential difference between this type and that without delays.     

时滞抛物方程组解的振动性定理

本文获得了一类时滞抛物方程组解振动的若干充分条件