Semigroups of Locally Lipschitzian Operators and applications

1.证明了R^n中半线性拉普拉斯方程径向解的存在性，并研究了解的渐近性态.具有非线性项f（u）=*u*^<p-1>u（*u*（）SY ≥. ()1)本文<q-1>研究了一个径向解，其中1&lt;p&lt;（n+2）/（n-2）&lt;q，并证明了任何径向解的性质都是 *chi*-&gt;*，类似于（i）c*chi*^-（n-2）&gt;或（ii）（）SY. ()c^&lt;**&gt;*chi*^&lt;-2/（q-1）&gt;.2。考虑了比上述f（u）更一般的非线性项，研究了对称区域上椭圆型方程的Dirichlet问题：环、球、球的外部和R^n.证明了在0（）SY ≤ 0（）中具有精确kappa零点的径向解的存在性。对于每个域和任意整数kappa（）SY ≥，证明了（）*chi*&lt;*。（）0.结果给出了非线性项存在径向解的一个弱充分条件.利用非线性局部Lipschitz算子半群的统一抽象理论证明了非线性Klein-Gordon方程、FitzHugh-Nagumo方程和二维Navier-Stokes方程弱解的存在性.引入了一类广义耗散算子，证明了该类算子的抽象Cauchy问题的差分近似解的存在性和收敛性。推广了已知的半群生成理论和常微分方程解的典型唯一存在性定理。

1. The existence of radial solutions for the semilinear Laplace equations in R^n is proved and the asymptotic behavior of the solutions is investigated. The elliptic equation with the nonlinear term f(u)=*u*^<p-1>u (*u*(〕SY.gtoreq.〔)1), =*u*^<q-1>u (*u*<1), where 1<p<(n+2)/(n-2)<q, is studied and it is shown that any radial solution behaves, as *chi*->*, like either (i)c*chi*^<-(n-2)> or (ii)(〕SY.+-.〔)c^<**>*chi*^<-2/(q-1)>.2. The more general nonlinear term than the above f(u) is considered and the Dirichlet problem of the elliptic equations in symmetric domains ; annulus, ball, exterior of ball and R^n are investigated. The existence of radial solution having exactly kappa zeros in 0(〕SY.ltoreq.〔)*chi*<* is proved for each domain and any integer kappa(〕SY.gtoreq.〔)0. The result gives a weak sufficient condision on the nonlinear term for the existence of radial solutions.3. The existence of weak solutions of nonlinear Klein-Gordon equations, FitzHugh-Nagumo equations and two dimensional Navier-Stokes equations is shown to be proved by using an unified abstract theory of semigroups of nonlinear locally Lip-schitzian operators.4. A class of generalized dissipative operators is introduced and the existence and the convergence of difference approximate solutions of abstract Cauchy problems for the operation in the class are shown. Both of the known theory ofgeneration of semigroups and the typical uniquely existence theorems of solutions of ordinary differential equations are extended.