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Decay estimates for the double dispersion equation with initial data in real Hardy spaces

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Abstract

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.

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Acknowledgements

The results for the linear problem in this paper are essentially contained in the master thesis of the second author, who has been a student at University of Bari.

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Correspondence to Marcello D’Abbicco.

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D’Abbicco, M., De Luca, A. Decay estimates for the double dispersion equation with initial data in real Hardy spaces. J. Pseudo-Differ. Oper. Appl. 11, 363–386 (2020). https://doi.org/10.1007/s11868-019-00287-1

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Keywords

  • Double dispersion equation
  • Cauchy problem
  • Fourier multiplier estimates
  • Real Hardy spaces
  • Decay estimates
  • Global small data solutions

Mathematics Subject Classification

  • 35L15
  • 35L30
  • 42B15
  • 42B30
  • 42B37