强阻尼粘弹性波动方程的通用吸引子的存在性

Existence of the Universal Attractor for a Strongly Damped Viscoelastic Wave Equation

设Ω为具有光滑边界的3的有界区域.对给定的ω≥0,考虑了如下具有强阻尼项的粘弹性波动方程:utt-ωΔut-k(0)Δu-∫∞0k'(s)φ(x)Δu(t-s)ds+φ(u)=f,x∈Ω,t≥0;u(x,0)=u0(x,0),ut(x,0)=/tu0(x,0),x∈Ω;u(x,t)=0,x∈Ω,t≥0.对非线性项施加非常一般的临界增长率的条件下,在能量空间X0=D(A12)×L2(Ω)×M1中证明了上述方程的通用吸引子的存在性.

Let Ω be a bounded domain in R3 with smooth boundary.Given ω≥0,w e consider the follwing strongly damped viscoelastic wave equation：utt-ωΔut-k（0）Δu-∫∞0k＇（s）φ（x）Δu（t-s）ds＋φ（u）=f,x∈Ω,t≥0;u（x,0）=u0（x,0）,ut（x,0）=/tu0（x,0）,x∈Ω;u（x,t）=0,x∈Ω,t≥0. we prove the existence of the universal attractor for the above equation,in the presence of a quite general nonlinearity of critical growth,on the energy space X0=D（A12）×L2（Ω）×M1.