非线性梁振动方程的周期解

PERIODIC SOLUTION OF NONLINEAR BEAM VIBRATION EQUATIONS

一、引言考虑下述问题Ku″+A^2u+M(‖A^1/2u‖~2)Au+Au′=f(x,t),t>0,x∈Ω,(1.1)u|_t=0~=u_0(x),x∈Ω,(1.2)Ku′|_(t=0)=u_1(x),x∈Ω,(1.3)u=0,x∈(?)Ω,t≥0 (1.4)的ω-周期解的存在性.其中 Ω(?)R^n 为一有界光滑区域,u′=((?)u)/((?)t),u_″=((?)u)/((?)t)~2,K 为有界线性对称算子且满足(Ku,u)≥0,M∈C^1[0,∞),M(ξ)≥-β,ξ≥0.此模型最初由Woinowsky 和 Krieger 提出。

This paper discusses the initial boundary value problem of the nonlinear hyperbolic equa-tions Ku″+A^2u+M(‖A^(1/2)+u‖~2)Au+Au′=F(x,t) from the beam vibration and proves that theproblem has w-periodic solution under some conditions about K,A,M and f.