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The Hp–Hq Estimates for a Class of Dispersive Equations and Related Applications

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Abstract

In this paper we study the Hp–Hq estimates of the solutions for a class of dispersive equations

$$\left\{ {\matrix{{i{\partial _t}u(t,x) = - P(|\nabla |)u(t,x),} \hfill & {(t,x) \in \mathbb{R} \times {\mathbb{R}^n},} \hfill \cr {u(0,x) = {u_0}(x),} \hfill & {x \in {\mathbb{R}^n},} \hfill \cr } } \right.$$

where P: ℝ+ → ℝ is smooth away from the origin and enjoy different scalings. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.

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Acknowledgements

The authors would like to express their deep gratitude to the referee for his/her very careful reading. Qingquan Deng was supported by the National Natural Science Foundation of China (No. 11971191).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis & Applications (Ministry of Education), Central China Normal University, Wuhan, 430079, China

    Qingquan Deng

  2. Department of Mathematical Science, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA

    Dashan Fan

  3. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Ruizhi Zhao

Authors
  1. Qingquan Deng
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  2. Dashan Fan
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  3. Ruizhi Zhao
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Correspondence to Qingquan Deng.

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Deng, Q., Fan, D. & Zhao, R. The Hp–Hq Estimates for a Class of Dispersive Equations and Related Applications. Front. Math (2024). https://doi.org/10.1007/s11464-023-0115-9

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  • DOI: https://doi.org/10.1007/s11464-023-0115-9

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