Capacity and Potential Estimates for Quasiminimizers

We obtain estimates for quasiminimizing potentials and their level sets. A new method, based on one dimensional quasiminimizers, is used. The connection of these estimates and the maximum principle is studied.     

Lipschitz Stability in Inverse Source Problems for Singular Parabolic Equations

We consider 2-D Klein-Gordon equation with quadratic nonlinearity and prove Strichartz type dispersive estimates for the global solution with small initial data in the Sobolev space H ^1+?.     

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of

$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$

with f subject to the general structural conditions

$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$

where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ L ^t (Ω) for some t > n/p ^?, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L ^∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L ^∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.

     

Stability of Solutions for a Class of Quasilinear Degenerate Parabolic Equations

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二阶拟线性退化抛物方程Cauchy问题解的稳定性

在一定的结构性条件下,证明了如下形式的拟线性退化抛物方程 Cauchy问题解的稳定性．     

Strong Stability and Non-smooth Data Error Estimates for Discretizations of Linear Parabolic Problems

We study smoothing properties of discretizations of a linear parabolic initial boundary value problem with a possibly non-selfadjoint elliptic operator. The solution at time t > 0 of this problem, as well as its time derivatives, are in L _r for initial values in L _s even when r > s. We show that similar strong stability results hold for discrete solutions obtained by discretizing in space by linear finite elements and in time by a class of A()-stable implicit rational multistep methods (including single step methods as a special case) with good smoothing properties, as well as for certain combinations of single step methods. Most of our results are derived from the corresponding L ₂-bounds, shown by semigroup techniques, together with a discrete Gagliardo-Nirenberg inequality, and generalize previously known estimates with respect to admissible problems and time discretization methods. Our techniques make it possible to obtain, e.g., supremum norm error estimates for initial data which are only required to be in L ₁.     

一类抛物型偏微分方程反问题的稳定性

本文讨论一类抛物型偏微分方程反问题，研究测量值在特定边界上给定时源项确定的稳定性，在合理的假设下证明了该反问题具有按Ｌｉｐｓｃｈｉｔｚ型连续依赖于测量值的稳定性，推广了Ｙａｍａｍｏｔｏ的结果．     

求解一般抛物方程侧边值问题的Fourier正则化方法

逆热传导问题是严重不适定问题,它的解如果存在,其解将不连续依赖于定解数据,使得数值计算和理论分析都非常困难。但目前关于逆热传导问题的已有文 献大都主要集中于讨论由标准热传导方程所描述的问题。该文给出了一种适用于由一般一维抛物方程所描述的逆热传导问题且具有Holder连续性的Fourier正则化新方法。     

四阶抛物方程一类变步长本性并行格式的稳定性分析

本文给出了四阶抛物方程一类变步长的本性并行差分格式,并用发展的能量法给出了该格式的绝对稳定性证明,得到了绝对稳定性估计.     

高维抛物型方程差分格式的稳定性

本文提出了一个改进抛物型方程差分格式稳定性条件的新方法,给出并证明新方法稳定的充要条件,数值例子显示了本方法的计算优越性.     

解抛物型方程的分支稳定的高精度显式差分格式

用待定参数法构造了解一维抛物型方程的分支稳定的高精度显式差分格式,截断误差为O（△t^4△x^4),稳定性条件为r=a△t/△x^2＜1/2。     

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

Let u _α be the solution of the It? stochastic parabolic Cauchy problem , where ξ is a space—time noise. We prove that u _α depends continuously on α , when the coefficients in L _α converge to those in L ₀ . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L _α tend to 0 the corresponding solutions u _α converge to the solution u ₀ of the degenerate Cauchy problem . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) ^* . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions. Accepted 22 May 1998     

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

一类非线性抛物型方程解的稳定性与唯一性

本文运用能量积分的方法，研究了一类非线性抛物型方程的解关于自由项与初始条件的稳定性与唯一性。     

Existence and Stability of Time-Periodic Dissipative Structures in Parabolic Systems of Reaction-Diffusion Type

On the interval 0 x the parabolic system

Stability in the Numerical Solution of Linear Parabolic Equations with a Delay Term

This paper is concerned with the stability of numerical processes that arise after semi-discretization of linear parabolic equations wit a delay term. These numerical processes are obtained by applying step-by-step methods to the resulting systems of ordinary delay differential equations. Under the assumption that the semi-discretization matrix is normal we establish upper bounds for the growth of errors in the numerical processes under consideration, and thus arrive at conclusions about their stability. More detailed upper bounds are obtained for -methods under the additional assumption that the eigenvalues of the semi-discretization matrix are real and negative. In particular, we derive contractivity properties in this case. Contractivity properties are also obtained for the -methods applied to the one-dimensional test equation with real coefficients and a delay term. Numerical experiments confirming the derived contractivity properties for parabolic equations with a delay term are presented.     

Parabolic transfer for real groups

We shall establish an identity between distributions on different real reductive groups. The distributions arise from the trace formula. They represent the main archimedean terms in both the invariant and stable forms of the trace formula. The identity will be an essential part of the comparison of these formulas. As such, it is expected to lead to reciprocity laws among automorphic representations on different groups.

Our techniques are analytic. We shall show that the difference of the two sides of the proposed identity is the solution of a homogeous boundary value problem. More precisely, we shall show that it satisfies a system of linear differential equations, that it obeys certain boundary conditions around the singular set, and that it is asymptotic to zero. We shall then show that any such solution vanishes.

     

Inverse Problems for Parabolic Equations

The determination of variable filtration through a layer is an important and actual problem in filtration theory. This problem has been examined in some particular cases, for instance in the papers [1, 2].