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Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions. (English) Zbl 1031.35124
The authors deal with the Cauchy problem for the GKdV equation \[ u_t+ u^p u_x+ u_{xxx}= 0,\quad u(x,0)= u_0(x),\quad t> 0,\quad x\in\mathbb{R} \] with the initial function \(u_0\) from a class of analytic functions in a symmetric strip around the real axis. The number \(p\) is a positive integer. T. Kato and K. Masuda [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 455-467 (1986; Zbl 0622.35066)] developed method of obtaining spatial analyticity of solutions for evolution equation. By applying it to the GKdV equation with analytic data they obtained that the solution will stay analytic in a strip the width of which may decrease as time advances.
The goal of this paper is to show that it is possible to obtain a local well-posedness (in Hadamard sense) of the Cauchy problem for a strip without shrinking the width of the strip in time. The prove is realized by Kato’s smoothing effect in the space \(C(\langle 0,T\rangle, G^{\sigma, s})\) where \(T\) is a positive time and \(G^{\sigma, s}\) is a Gevrey space for \(s\geq 0\) if \(p> 1\) and \(s> 3/2\) if \(p\geq 2\) and \(\sigma> 0\).

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)