Hayashi, Nakao Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and the Szegö spaces on a sector. (English) Zbl 0729.35119 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 Cited in 10 Documents 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 PDF BibTeX XML Cite \textit{N. Hayashi}, Duke Math. J. 62, No. 3, 575--591 (1991; Zbl 0729.35119) Full Text: DOI References: [1] H. Aikawa, N. Hayashi, and S. Saitoh, The Bergman space on a sector and the heat equation , Complex Variables Theory Appl. 15 (1990), no. 1, 27-36. · Zbl 0704.30012 [2] H. Aikawa, Isometrical identities for the Bergman and the Szegö spaces on a sector , J. Math. Soc. Japan, · Zbl 0728.30004 · doi:10.2969/jmsj/04310195 [3] J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation , SIAM J. Math. Anal. 20 (1989), no. 6, 1388-1425. · Zbl 0702.35224 · doi:10.1137/0520091 [4] N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation , · Zbl 0742.35056 · doi:10.1137/0522107 [5] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation , Studies in applied mathematics, Advances in Mathematics Supplememtary Studies, vol. 8, Academic Press, New York, 1983, pp. 93-128. · Zbl 0549.34001 [6] T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. I , Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 6, 455-467. · Zbl 0622.35066 · numdam:AIHPC_1986__3_6_455_0 · eudml:78123 [7] C. E. Kenig, G. Ponce, and L. Vega, On the (generalized) Korteweg-de Vries equation , Duke Math. J. 59 (1989), no. 3, 585-610. · Zbl 0795.35105 · doi:10.1215/S0012-7094-89-05927-9 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.