具有时间依赖系数的波方程拉回吸引子的存在性

Existence of pullback attractors for wave equations with time dependent coefficients

摘要

研究具有时间依赖系数的非线性波方程。首先,根据一致扇形算子理论证明解的适定性。其次,通过构造恰当的泛函获得过程的有界耗散性。最后,利用压缩函数的方法证明过程的渐近紧性,从而得到其拉回吸引子的存在性。在问题的研究中,由于将时间依赖系数分解为正部与负部,因此在证明过程的渐近紧性时,无法应用算子分解技巧,从而选择了压缩函数的方法。

Abstract

This paper is concerned with the nonlinear wave equations with time-dependent coefficients. Firstly, the well-posedness of the solution is proved based on the theory of the uniform sector operator. Secondly, the bounded dissipation of the process is obtained by constructing appropriate functionals. Finally, the asymptotic compactness of the process is proved by using the method of the compression function. In the research of the problem, since the time-dependent coefficients α₁ are decomposed into positive and negative parts, the operator decomposition technique cannot be applied when proving the asymptotic compactness of the process. Therefore, the method of compressing the function is chosen.

关键词

波方程 / 适定性 / 拉回吸引子 / 时间依赖系数 / 耗散性 / 紧性

Key words

contractive function / well-posedness / pullback attractors / time-dependent coefficients / dissipation / compactness

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