Nondecreasing analytic radius for the a Kawahara-Korteweg-de-Vries equation

Aissa Boukarou; Khaled Zennir; Mohamed Bouye; Abdelkader Moumen; Aissa Boukarou; Khaled Zennir; Mohamed Bouye; Abdelkader Moumen

doi:10.3934/math.20241090

AIMS Mathematics

2024, Volume 9, Issue 8: 22414-22434. doi: 10.3934/math.20241090

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Nondecreasing analytic radius for the a Kawahara-Korteweg-de-Vries equation

1.
Faculty of Mathematics, University of Science and Technology Houari Boumediene, Bab Ezzouar, Algeria; Email: boukarouaissa@gmail.com
2.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia; Email: k.zennir@qu.edu.sa
3.
Department of Mathematics, Faculty of Science, University 20 Août 1955- Skikda, Algeria; Email: k.zennir@univ-skikda.dz
4.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia; Email: mbmahmad@kku.edu.sa
5.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia; Email: mo.abdelkader@uoh.edu.sa

Received: 28 May 2024 Revised: 07 July 2024 Accepted: 10 July 2024 Published: 18 July 2024
MSC : 35G25, 35K55

By using linear, bilinear, and trilinear estimates in Bourgain-type spaces and analytic spaces, the local well-posedness of the Cauchy problem for the a Kawahara-Korteweg-de-Vries equation
$ \partial_{t}u+\omega\partial_{x}^{5}u+\nu \partial_{x}^{3}u+\mu\partial_{x}u^{2}+\lambda\partial_{x}u^{3}+\mathfrak{d}(x)u = 0, $
was established for analytic initial data $ u_{0} $. Besides, based on the obtained local result, together with an analytic approximate conservation law, we prove that the global solutions exist. Furthermore, the analytic radius has a fixed positive lower bound uniformly for all time.
- analytic radius,
- Kawahara-Korteweg-de-Vries equation,
- damped effect,
- nonlinear equations
Citation: Aissa Boukarou, Khaled Zennir, Mohamed Bouye, Abdelkader Moumen. Nondecreasing analytic radius for the a Kawahara-Korteweg-de-Vries equation[J]. AIMS Mathematics, 2024, 9(8): 22414-22434. doi: 10.3934/math.20241090

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Abstract

By using linear, bilinear, and trilinear estimates in Bourgain-type spaces and analytic spaces, the local well-posedness of the Cauchy problem for the a Kawahara-Korteweg-de-Vries equation

$ \partial_{t}u+\omega\partial_{x}^{5}u+\nu \partial_{x}^{3}u+\mu\partial_{x}u^{2}+\lambda\partial_{x}u^{3}+\mathfrak{d}(x)u = 0, $

was established for analytic initial data $ u_{0} $. Besides, based on the obtained local result, together with an analytic approximate conservation law, we prove that the global solutions exist. Furthermore, the analytic radius has a fixed positive lower bound uniformly for all time.

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