具结构阻尼的拟线性膜方程的吸引子及其上半连续性

Attractors and their upper semicontinuity for the quasi-linear membrane equation with structural damping

本文研究具结构阻尼的拟线性膜方程utt+Δ^(2)u+(-Δ)^(α)ut+Δφ(Δu)+f(u)=g的适定性以及解的长时间动力学行为,其中α∈(1,2),旨在研究耗散指标α对方程解的适定性和长时间动力学行为的影响.本文证明非线性项φ(s)存在一个依赖于耗散指标α的临界指数pα=(N+4(α-1))/(N-4(α-1))^(+)(N=3,4),当1≤p<p_(α)时,对f(u)没有任何多项式增长限制:(ⅰ)方程的初边值问题是适定的,其解当t> 0时具有整体正则性;(ⅱ)对任意α∈(1,2),对应的解算子半群S^(α)(t)在自然能量空间中存在整体吸引子和指数吸引子;(ⅲ)整体吸引子族{Aα}在任意点α0∈(1,2)处上半连续,即对Aα0。的任意邻域U,当|α-α0|<<1时有Aα■U.

The paper investigates the well-posedness and longtime dynamics of the quasi-linear membrane equation with structural damping:utt+Δ^(2)u+(-Δ)^(α)ut+Δφ(Δu)+f(u)=g,withα∈(1,2).We are concerned with the influence of the dissipative indexαto the well-posedness and longtime dynamics of the equation.We show that there exists a critical exponent pα=(N+4(α-1))/(N-4(α-1))^(+)of the nonlinearityφ(s),which depends on the dissipative indexα,such that when 1≤p<pα,without any polynomial growth assumption on the nonlinearity f(u)for the cases:either N=3 or 1<p<pα,N=4 and without any polynomial growth restriction on f(u)for the case p=1,N=4,(i)the initial boundary value problem of the equation is well-posed and its solution possesses additionally global regularity;(ii)the related solution semigroup Sα(t)has a global and an exponential attractor in natural energy space for eachα∈(1,2),respectively;(iii)the family of global attractors{Aα}is upper semicontinuous at each pointα0∈(1,2),i.e.,for any neighborhood U of Aα0,Aα■U when|α-α0|≤1.