变系数Euler-Bernoulli梁振动发展系统的存在性

摘要

讨论变系数Euler-Bernoulli梁振动系统utt(x,t)+η(t)uxxxx(x,t)=0,0＜x＜1,0≤t≤T{ u(0,t)=ux(0,t)=0,0≤t≤T-uxxx(1,t)+mutt(1,t)=-αut(1,t)+βuxxxt(1,t),0≤t≤T (1)uxt(1,t)=-γuxx(1,t),0≤t≤Tu(x,0)=u1(x),ut(x,0)=u2(x),0≤x≤1证明了该系统产生一个发展系统.%A flexible structure consisting of Euler-Bernoulli beam with a variable coefficient described by utt(x,t) +-η(t)uxxxx (x,t)=0, 0＜x＜1, 0≤t≤T u(0,t) = ux(0,t) = 0, 0 ≤ t ≤ T { - uxxx(1,t) +mutt(1,t) =- aut(1,t) + βuxxxt(1,t), 0≤t≤T (1) uxt(1,t) =-γuxx(1,t), 0≤t≤T u(x,0) =u1(x), ut(x,0) =u2(x), 0≤x≤ 1 rnis considered. We prove that the system generate an evolution system when η(t) is a continuous function and η(t) ∈∨ [0,T], 0＜η0≤η(t)≤M.