Enhanced existence time of solutions to evolution equations of Whitham type

Mats Ehrnström; Yuexun Wang

doi:10.3934/dcds.2022035

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2022, Volume 42, Issue 8: 3841-3860. Doi: 10.3934/dcds.2022035

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Enhanced existence time of solutions to evolution equations of Whitham type

Mats Ehrnström^1, and
Yuexun Wang^2,3, ,

1.
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
2.
School of Mathematics and Statistics, Lanzhou University, 730000 Lanzhou, China
3.
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France

^* Corresponding author
^* Corresponding author

Both authors acknowledge the support by grant nos. 231668 and 250070 from the Research Council of Norway. The second author also acknowledges the support of the ANR project ANuI, and the partial support of grant no. 830018 from China

Received: July 2021

Revised: February 2022

Early access: April 2022

Published: August 2022

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Abstract

We show that Whitham type equations $u_t + u u_x -\mathcal{L} u_x = 0$, where $L$ is a general Fourier multiplier operator of order $\alpha \in [-1, 1]$, $\alpha\neq 0$, allow for small solutions to be extended beyond their ordinary existence time. The result is valid for a range of quadratic dispersive equations with inhomogenous symbols in the dispersive regime given by the parameter $\alpha$.

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Mathematics Subject Classification: Primary: 76B15, 76B03; Secondary: 35S30, 35A20.

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Figure 1. The symmetries of the function $\varphi$

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