Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and the Szegö spaces on a sector. (English) Zbl 0729.35119

Hayashi, Nakao

Duke Math. J. 62, No. 3, 575-591 (1991).

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

Keywords:

Bergman space; Szegö space; Korteweg-de Vries equation; unique local solution

References:

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