In this paper we prove that every compact invariant subset $$\mathscr{A}$$ associated with the semigroup S n,k ( t ) t ≥0 generated by wave equations with variable damping, either in the interior or on the boundary of the domain where Ω ⊂ ℝ 3 is a smooth bounded domain, in H 0 1 (Ω) × L 2 (Ω) is in fact bounded in D ( B 0 ) × H 0 1 (Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H 1 (Ω) × L 2 (Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.

中文翻译：

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可变内阻尼波方程中不变量集的正则性

在本文中，我们证明了每个与半群 S n,k ( t ) t ≥0 相关联的紧致不变子集 $$\mathscr{A}$$ 由具有可变阻尼的波动方程生成，无论是在内部还是在边界上Ω ⊂ ℝ 3 是光滑有界域，在 H 0 1 (Ω) × L 2 (Ω) 中的域实际上在 D ( B 0 ) × H 0 1 (Ω) 中有界。作为我们的结果的应用，当内部变量阻尼收敛到边界阻尼时，我们以 H 1 (Ω) × L 2 (Ω) 的范数获得弱阻尼半线性波动方程的全局吸引子的上半连续性。分布感。