We study the Cauchy problem for the linear double dispersion equation$$\begin{aligned} u_{tt}-\Delta u_{tt}+\Delta ^2 u-\Delta u-\Delta u_t =0, \quad t\ge 0,\ x\in {\mathbb {R}}^n \end{aligned}$$and we derive long time decay estimates for the solution in \(L^p\) spaces and in real Hardy spaces. We employ the obtained results to study the equation with nonlinearity \(\Delta f(u)\) and nonsmooth f.

中文翻译：

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实数Hardy空间中具有初始数据的双色散方程的衰减估计

我们针对线性双色散方程$$ \ begin {aligned} u_ {tt}-\ Delta u_ {tt} + \ Delta ^ 2 u- \ Delta u- \ Delta u_t = 0，\ quad t \ ge 0，\ x \ in {\ mathbb {R}} ^ n \ end {aligned} $$中，我们得出\（L ^ p \）空间和实际Hardy空间中解的长时间衰减估计 。我们使用获得的结果来研究具有非线性\（\ Delta f（u）\）和非光滑 f的方程 。