A generalization of a result of G. Pólya and its application to a continuous extension of the de la Vallée Poussin means

Variation diminishing properties are established for the periodic kernels . On the real axis, there are related variation diminishing properties of the functions u^msgnu, which are the Green’s functions for the differential operator D^(m+1).     

非光滑系数抛物方程弱解的双倍性质和唯一延拓性

对带奇异位势的非光滑系数抛物方程ut-div(A▽xu) Vu=0进行了讨论,其中A满足一致椭圆条件和Dini连续性,V是Kato型奇异位势．建立了上述抛物方程弱解u的双倍测度性质以及唯一延拓性定理．     

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.     

On the sign-stability of numerical solutions of one-dimensional parabolic problems

The preservation of the qualitative properties of physical phenomena in numerical models of these phenomena is an important requirement in scientific computations. In this paper, the numerical solutions of a one-dimensional linear parabolic problem are analysed. The problem can be considered as a altitudinal part of a split air pollution transport model or a heat conduction equation with a linear source term. The paper is focussed on the so-called sign-stability property, which reflects the fact that the number of the spatial sign changes of the solution does not grow in time. We give sufficient conditions that guarantee the sign-stability both for the finite difference and the finite element methods.     

Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity

The Cauchy problem

(P)     

具有边界摄动的抛物型方程的可解性

本文研究了具有边界摄动的抛物型方程.利用可解性的讨论,得到了原问题的摄动解.     

中立型抛物方程振动的一个充分必要条件

本文得到了中立型抛物方程振动一个充分必要条件．     

Complete quenching for singular parabolic problems

We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u₀(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t₀>0 such that the solution is identically equal to zero.     

Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems

In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on R. These results are not restricted to positive solutions.     

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).     

APPLICATION OF THE G CLASS OF FUNCTIONS IN THE PARABOLIC CLASS

§1　IntroductionIn[1],wehaveintroducedtheconceptoftheGclassoffunctionsintheparabolicclass,andhaveprovedtheHldercontinuityofthiskindoffunctions.Theintroductionoftheconceptcontributestotheproofoftheregularityandexistenceofthesolutionforthefirstboundaryvalueproblemofparabolicequationindivergenceform.Here,weconsidertheapplicationsoftheGclassoffunctionsintheparabolicclasstothefirstboundaryvalueproblemofparabolicequation.Asweknow,ithasreceivedextensivestudyforthefirstboundaryvalueproblemofthefoll…     

一类拟线性抛物方程的均匀化

本文研究具有振荡系数的二阶抛物方程δｔｂ（ｕ）－δｘｉ（αｉｊ（ｘ／ε）δｘｊｕ＋αｉ（ｂ（ｕ）））＝ｆ（ｂ（ｕ））的渐近性态。由于函数ｂ（ｕ）的导函数ｂ‘（ｕ）允许为非线性函数，并可在ｕ＝０处退化，研究了均匀化问题通常的能量方法在这里行不通。     

Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u₀(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P^∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.     

Oscillation of the solutions of parabolic equations with nonlinear neutral terms

In the present paper the oscillatory properties of the solutions of parabolic equations with nonlinear neutral terms are investigated. Our approach is to reduce the multi-dimensional problem to a one-dimensional problem for delay differential inequalities.     

时滞抛物方程解振动的充要条件

In this paper, necessary and sufficient conditions for oscillation of solu-tions of delay parabolic equations are obtained.     

A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.