Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices
Tóm tắt
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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