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Decay of solutions to a water wave model with a nonlocal viscous term

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Abstract

We update here the results on the decay of solutions to a nonlocal water wave equation that reads

$$\begin{aligned} u _t+u_x+\dfrac{1}{\sqrt{\pi }}\frac{\partial }{\partial t} \int _0^t\dfrac{u(s)}{\sqrt{t-s}} ds+ u u_x= u_{xx}, \end{aligned}$$

where \(\displaystyle \dfrac{1}{\sqrt{\pi }}\dfrac{\partial }{\partial t} \int _0^t\dfrac{u(s)}{\sqrt{t-s}} ds \) is the Riemann–Liouville half-order derivative.

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Acknowledgements

These result were presented at the PACOM 2017 conference. The authors would like to thank the anonymous referees for their remarks and suggestions.

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Authors and Affiliations

  1. Université de Nîmes and IMAG CNRS 5149, Nîmes, France

    S. Dumont

  2. LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039, Amiens, France

    O. Goubet

  3. Unité de recherche: Multifractales et Ondelettes, Faculté des Sciences de Monastir, Av. de l’environnement, 5019, Monastir, Tunisia

    I. Manoubi

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  1. S. Dumont
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Correspondence to O. Goubet.

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Dumont, S., Goubet, O. & Manoubi, I. Decay of solutions to a water wave model with a nonlocal viscous term. Afr. Mat. 31, 115–127 (2020). https://doi.org/10.1007/s13370-019-00748-2

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