具有结构阻尼的Kirchhoff型波方程的时间依赖全局吸引子

The Time-Dependent Global Attractors for Kirchhoff-Type Wave Equation with Structural Damping

本文讨论了具有结构阻尼的Kirchhoff型波方程:ε(t)■_(t)^(2)-M(||■u||^(2))Δu+(-Δ)^(γ)■_(t)u+f(u)=g(x).γ∈(1/2,1)解的适定性和长时间行为.当非线性项f的增长指数满足2≤p≤3+2γ时,借助Faedo-Galerkin逼近方法和渐近正则估计,得到了解的适定性和正则性.继而利用收缩函数方法验证解过程的渐近紧性.最终证明了时间依赖全局吸引子在自然能量空间H_(0)^(1)(Ω)×L^(2)(Ω)中的的存在性.

In this paper,the authors study the well-posedness and the longtime dynamics of the solutions to the Kirchhoff wave equation with structural damping:ε(t)■_(t)^(2)-M(||■u||^(2))Δu+(-Δ)^(γ)■_(t)u+f(u)=g(x).whereγ∈(1/2,1)When the growth exponent satisfies 1≤p≤3+2γfirstly,by use of Faedo-Galerkin approximation method and asymptotically regular estimate technique,the well-posedness and regularity of solutions are established.Secondly,the asymptotic compactness of the solution process is proved via the method of contraction function.Finally,the existence of time-dependent global attractor is obtained in the natural energy space H_(0)^(1)(Ω)×L^(2)(Ω)。