Upper Estimate of the Interval of Existence of Solutions of a Nonlinear Timoshenko Equation

Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t-interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.     

First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

In Ref. 1, bilevel programming problems have been investigated using an equivalent formulation by use of the optimal value function of the lower level problem. In this comment, it is shown that Ref. 1 contains two incorrect results: in Proposition 2.1, upper semicontinuity instead of lower semicontinuity has to be used for guaranteeing existence of optimal solutions; in Theorem 5.1, the assumption that the abnormal part of the directional derivative of the optimal value function reduces to zero has to be replaced by the demand that a nonzero abnormal Lagrange multiplier does not exist.     

一种求二元有理插值函数的方法

给出一种方法可直接计算基于矩形节点的二元有理插值函数的分母在节点处的值 ,进而判断相应的二元有理插值函数是否存在 .此方法运用灵活 ,适用范围广 ,在相应的有理插值函数存在时 ,能给出它的具体表达式 .此外 ,我们还针对文中两个主要逆矩阵 ,给出了相应的递推公式 ,避免了求逆计算 .     

非局部非线性发展方程解的存在性与唯一性

本文利用Ｂａｎａｃｈ不动点定理证明一类非线性发展方程的非局部Ｃａｕｃｈｙ问题解的存在性与唯一性，及其对非局部拟线性抛物方程的应用。     

SOME RESULTS ON A DEGENERATE AND SINGULAR DIFFUSION EQUATION

In this article,a degenerate and singular diffusion problem is studied.The ex- istence of solutions is established by parabolic regularization.Some properties of solutions, for instance,asymptotic behavior,are also discussed.     

Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equation with periodic boundary conditions

In the hyper-viscous Navier-Stokes equations of incompressible flow, the operator A=−Δ is replaced by A_α,a,b≡aA^α+bA for real numbers α,a,b with α?1 and b?0. We treat here the case a>0 and equip A (and hence A_α,a,b) with periodic boundary conditions over a rectangular solid Ω⊂Rn. For initial data in Lp(Ω) with α?n/(2p)+1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for α=1 and b=0. Specializing to the case p=2, which holds a particular physical relevance in terms of the total energy of the system, it is somewhat interesting to note that the condition α?n/4+1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case α=n/4+1/2 we generalize standard existing (for n=3) “folklore” results and use energy techniques and Gronwall's inequality to obtain first a time-dependent H^α-bound, and then convert to a time-independent global exponential H^α-bound. This is to be expected, given that uniform bounds already exist for n=2,α=1 ([6, pp. 78-79]), and the folklore bounds already suggest that the α?n/4+1/2 cases for n?3 should behave as well as the n=2 case. What is slightly less expected is that the n?3 cases are easier to prove and give better bounds, e.g. the uniform bound for n?3 depends on the square of the data in the exponential rather than the fourth power for n=2. More significantly, for α>n/4+1/2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform L²-estimate and are algebraic in terms of the uniform L²-bounds on the initial and forcing data. The integer powers on the square of the data increase without bound as α↓n/4+1/2, thus “anticipating” the exponential bound in the borderline case α=n/4+1/2. We prove our results for the case a=1 and b=0; the general case with a>0 and b?0 can be recovered by using norm-equivalence. We note that the hyperviscous Navier-Stokes equations have both physical and numerical application.     

Solutions globales d'un problème de Cauchy linéaire

On résoud le problème de Cauchy pour des opérateurs aux dérivées partielles à coefficients polynomiaux par rapport aux variables d'espace dans des classes de Gevrey projectives. Nos résultats sont des versions globales du théorème de Cauchy-Kowalewski et du théorème de Nagumo.