Abstract
Consider the generalized dispersive equation defined by
where \(\phi(\sqrt{-\Delta})\)s is a pseudo-differential operator with symbol φ(|ξ|). In the present paper, assuming that φ satisfies suitable growth conditions and the initial data in Hs(ℝn), we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations (*) fails. These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.
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Acknowledgements
The authors would like to express their deep gratitude to the referees for their very careful reading, important comments and valuable suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11661061, 11761054), the Inner Mongolia University Scientific Research Projects (No. NJZY19186), and the Natural Science Foundation of Inner Mongolia (No. 2019MS01003).
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Lan, S., Li, T. & Niu, Y. Dimension of divergence sets for dispersive equation. Front. Math. China 15, 317–331 (2020). https://doi.org/10.1007/s11464-020-0835-z
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DOI: https://doi.org/10.1007/s11464-020-0835-z