图上的非线性克莱因—戈尔登方程

设G=(V,E)是有限连通权重图,Ω是V的有限子集,且满足Ω°≠?.本文讨论了有限图上的非线性克莱因—戈尔登方程{∂^(2)_(t)u−ΔΩu+m^(2)u+|u|ρ−1⋅u=0,(t,x)∈[0,∞)×Ω∘,u(0,x)=g(x),x∈Ω∘,∂_(t)u(0,x)=h(x),x∈Ω∘,u(t,x)=0,(t,x)∈[0,∞)×∂Ω,其中△Ω是狄利克雷—拉普拉斯算子,m和ρ是[1,∞)内的常数.通过使用[Math.Ann.,1930,102(1):650-670]中介绍的Rothe方法,在一定条件下证明了上述方程解的存在性,并给出了ρ=1时该方程的显式解.

DIRECT CONSTRUCT OF PERIODIC SOLUTIONS TO PERTURBED KLEIN-GORDON EQUATIONS BY NEWTON ITERATION

By introducing the block estimate technique and directly using the Newton iteration method, the author constructs Cantor families of time periodic solutions to a class of nonlinear wave equations with periodic boundary conditions. The Lyapunov-Schmidt decomposition used by J. Bourgain, W. Craig and C. E. Wayne is avoided. Thus this work simplifies their framework for KAM theory for PDEs.