Abstract
We first prove that the Cauchy problem of the Kawahara equation, \( \partial _{t} u + u\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0 \), is locally solvable if the initial data belong to H r(R) and \( r \geqslant - \frac{7} {5} \) , thus improving the known local well-posedness result of this equation. Next we use this local result and the method of "almost conservation law" to prove that global solutions exist if the initial data belong to H r(R) and \( r > - \frac{1} {2}. \)
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This work is partially supported by the China National Natural Science Foundation (No.10471157), and the second author is also supported in part by the Advanced Research Center of the Sun Yat-Sen University
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Wang, H., Cui, S.B. & Deng, D.G. Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices. Acta Math Sinica 23, 1435–1446 (2007). https://doi.org/10.1007/s10114-007-0959-z
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DOI: https://doi.org/10.1007/s10114-007-0959-z