In this paper, we study the Cauchy problem for the generalized double dispersion equation with structural damping. The equation behaves as the usual diffusion phenomenon over the low frequency domain, while it admits a feature of regularity-loss on the high frequency part. The feature of regularity-loss leads to the weakly dissipative property of the equation. To overcome the weakly dissipative property, the time-weighted energy is introduced, and extra regularity on the initial data is required. Under suitable conditions on the initial data and space dimensions, we prove the global existence and time-decay rates of solutions. The proof is based on the spectral analysis for the solution operators, time-weighted energy, and the contraction mapping theorem. Moreover, we also establish the asymptotic profiles of global solutions involving the nonlinear term for n ≥ 3, ν∈(0,12) and n ≥ 4, ν∈[12,1), respectively.

中文翻译：

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具有结构阻尼的广义双色散方程解的渐近线

在本文中，我们研究了具有结构阻尼的广义双色散方程的柯西问题。该方程在低频域表现为通常的扩散现象，但在高频部分具有规律性损失的特征。正则性损失的特征导致方程的弱耗散特性。为了克服弱耗散特性，引入了时间加权能量，并且需要对初始数据具有额外的规律性。在初始数据和空间维度的合适条件下，我们证明了解的全局存在性和时间衰减率。证明基于解算符的谱分析、时间加权能量和收缩映射定理。而且，