Semigroups of Locally Lipschitzian Operators and applications

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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.