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Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and the Szegö spaces on a sector. (English) Zbl 0729.35119
Let $$\Delta =\{z:$$ $$-\alpha <\arg z<\alpha$$, $$0<\alpha <\pi /3\}\cup \{z:\pi -\beta <\arg z<\pi +\beta$$, $$\pi /3<\beta <\pi /2\}$$. In the Bergman and the Szegö spaces on $$\Delta$$ we consider the (generalized) Korteweg-de Vries equation: $(*)\quad \partial_ tu+\partial^ 3_ xu+a(u)\partial_ xu=0,\quad (t,x)\in {\mathbb{R}}^+\times {\mathbb{R}};\quad u(0,x)=\phi (x),\quad x\in {\mathbb{R}},$ where $$a(\lambda)=\lambda^ p$$, $$p\in {\mathbb{N}}$$ and $$\phi$$ is a complex valued function. We assume that $$\phi \in H^{\infty}({\mathbb{R}})$$ $$(\equiv \cap H^ s({\mathbb{R}}))$$ has an analytic continuation $$\Phi$$ belonging to the Bergman and the Szegö spaces on $$\Delta$$. Then we show that (*) has a unique local solution u which has an analytic continuation U on $$\Delta$$ and U belongs to the same function spaces as does the initial data $$\Phi$$.
Reviewer: Nakao Hayashi

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35B60 Continuation and prolongation of solutions to PDEs
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##### References:
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