Abstract
In this paper we study solvability of the Cauchy problem of the Kawahara equation \( \partial _{t} u + au\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0 \) with L 2 initial data. By working on the Bourgain space X r,s(R 2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H r(R) and −1 < r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L 2(R).
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Project supported by the China National Natural Science Foundation (Grants 10171111, 10171112)
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Cui, S.B., Deng, D.G. & Tao, S.P. Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data. Acta Math Sinica 22, 1457–1466 (2006). https://doi.org/10.1007/s10114-005-0710-6
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DOI: https://doi.org/10.1007/s10114-005-0710-6