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Journal of Applied Nonlinear Dynamics 13(4) (2024) 619--630 | DOI:10.5890/JAND.2024.12.001

Aissa Boukarou$^1$, Kaddour Guerbati$^2$, Khaled Zennir$^{3,4}$, Aouatef Mansouri$^5$

$^1$ Dynamic Systems Laboratory, Department of Mathematics, University of Science and Technology Houari Boumediene, Algiers 16111, Algeria

$^2$ Laboratoire de Math'ematiques et Sciences appliqu'ees Universit'e de Ghardaia, Alg'erie

$^3$ Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia

$^4$ Department of mathematics, Faculty of sciences, University 20 A^out 1955- Skikda, Algeria

$^5$ Universit'e Larbi Ben Mhidi, Oum El Bouaghi, Alg'erie

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Abstract

In this article, using multi-linear estimate in Bourgain type spaces, we prove the local well-posedness of initial value problem associated with the equation $\partial_{t}w +\partial_{x}^{3}w +\eta(t) \mathcal{L}w+\partial_{x}(w)^{k} = 0,$ $ k=2, 4 $. The solution is established on the line for analytic initial data $u_{0}$ that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improve earlier results in [1].

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