半线性的

The paper first obtains the L2 -apriori estimates for the solutions of two kinds of autocatalytic model s under Dirichlet boundary conditions, and then, using the properties of linear semigroup and delicate calculations, the estimates of the maximal norms are obtained, therefore, the global existence of the solutions is proved.

该文利用重要不等式及能量积分方法首先得到了解的初等的先验估计。然后利用线性半群的有关性质及精细计算得到了解的最大模估计，从而证明了两类三次自催化模型在Dirichlet边界条件下整体解的存在性，并进而证明了第一类模型的最大吸引子的存在性

The maximum principles for functions which are defined on solutions of semilinear elliptic equations u+f=0(q=|u|~2)subject to Dirichlet boundary conditions u=0 were studied by using Hopf s maximum principle,the estimate of gradient q was obtained.

运用Hopf 极值原理讨论了一类具有Dirichlet边界条件u=0的半线性椭圆方程Δu+f=0(q=|u|2)的解的某个函数的极值原理，利用该结论获得了解的梯度q的估计。

Apparently, the approximate controllability doesn"t hold for generally linear parabolic equations with bounded control. It doesn"t hold for the semilinear heat equations presented there,either. Then, it is proved by analyzing its" solutions that no matter how long the time T is , there are targets that can"t be approximated by the solution of heat equation with Dirichlet boundary conditions;If a given target satisying certain conditions can be reached by the solution of heat equation .

本文先针对两个具体的半线性热方程系统证明了控制加约束后系统是不能控的；然后通过分析方程的解，对带位势及Dirichlet边界条件的热方程系统，无论时间多大，都存在不能控的目标；文中还证明了满足一定条件的目标可达。

The numercial simulation has been carried out, and the experimental results show that this method has been improved significantly comparing with the traditional one in nonlinear optics. Secondly, a second order system of semilinear elliptic equations, which comes from a mathematics model of the electric potential distribution of a media surrounded with the protein, is transformed into a variational problem, the existence of solution of it is proved by using variational principle and Trudinger—Moser inequality, and the relation between the solutions of variational problem and the solutions of its dual problem is obtained.

首先，讨论了非线性光学中的二次谐波产生的耦合方程组，利用变分法证明了耦合方程组非平凡解的存在性，然后进行了数值模拟，实验结果表明文中方法比经典的非线性光学中的方法有较大的改进，这对优化光倍频器件的设计将有所帮助；其次，将一类二阶半线性椭圆型方程组转化为一个变分问题，然后利用变分法和

Inthe first section,under the non-convex case we are devoted to investigate an optimalcontrol problem for a class of semilinear elliptic type equations which admit exactly twosolutions.By analyzing the property of a soltuion of the state equation and establishing the proper energy estimate,we prove the relaxed control method is fit for our problemand gain the Pontryagin\'s maximum principle.

第一节，在非凸情形下我们致力于一类只有两个解的半线性椭圆型方程的最优控制问题，通过对状态方程解性质的分析并建立适当的能量估计，我们证明了松弛控制方法对本问题的适用性并得到了最优对的Pontryagin\'s最大值原理。

A L(superscript ∞) bound estimate is provided on the solutions of a semi-linear parabolic system by the theory of phase transitions. The existence of solutions of corresponding hyperbolic system of conservation laws is studied by viscosity methods and compensated compactness framework.

讨论来源于物理相变理论的一类半线性抛物型方程组解的一致L估计，该结果是利用粘性法和补偿列紧框架研究相应的双曲型守恒律解的存在性的重要内容。

By constructing suitable cone and applying the decomposition technique of equation, the positive solutions are investigated for a class of semilinear fourth-order boundary value problems with first derivative.

通过构造适当的锥并且利用方程的分解技巧研究了一类含一阶导数的半线性四阶边值问题的正解。

Based on the surjection of maximal monotone operators, we prove that some semilinear heat equations with weak Lipschitz nonlinear terms are exactly controllable by L〓 0, T: H〓(Ω control; Based on the critical point theorem of coercive convex functionals, we prove that some parabolic equations with Lipschitz nonlinear terms are globally approximately controllable and finite-dimensional exactly controllable by controls acting on mobile supports, and finally, we extend the results to parabolic equations in R〓.

首先利用极大单调算子的满射原理证明了半线性热方程在较弱Lipschitz条件下可通过L〓0.T；H〓(Ω控制实现精确能控。然后通过构造强制凸泛函并利用其临界点理论证明了带Lipschitz非线性项的抛物型方程可通过作用在在移动支集上的控制来实现整体近似能控与有限维精确能控，最后将整体近似能控与有限维精确能控结论推广到无界区域R〓上。

In the last part, we study the existence of the solutions for semilinear subelliptic equations and the estimates of the eigenvalues for subelliptic operators on H.

另外我们还研究了Heisenberg群上一类半线性次调和方程的解的存在性问题及Heisenberg上群一类次调和算子的特征值估计问题。

We also obtain different conditions for the problem whether the semi-linear differential equation has solution according to different homotopy deformations.

根据同伦变换的不同形式，又得到了半线性微分方程不同的有解的条件。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新