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Nonlinear evolution equations and analyticity. I. (English) Zbl 0622.35066
Under an abstract framework, the authors first prove a theorem for a nonlinear evolution equation of form \(\partial_ tu=F(t,u)\) in a Banach space. The theorem gives an estimate for a certain family of Lyapunov functions for the solutions of the equation. Then, using the abstract result, the authors are able to prove that for Korteweg-de Vries and Benjamin-Ono equations, if the initial states have analytic continuation which are analytic and \(L^ 2\) in a strip containing the real axis, then the solutions have the same property for all time, but, in general, the width of the strip might be decreasing as time goes to infinity.
Reviewer: J.Yong

35Q99 Partial differential equations of mathematical physics and other areas of application
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35G20 Nonlinear higher-order PDEs
35A20 Analyticity in context of PDEs
Full Text: DOI Numdam EuDML
[1] Bourbaki, N., Fonctions d’une variable réelle, (1949), Hermann Paris · Zbl 0036.16801
[2] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., t. 32, 121-251, (1979) · Zbl 0388.34005
[3] Fritz, J.; Dobrushin, R. L., Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction, Comm. Math. Phys., t. 57, 67-81, (1977) · Zbl 0987.82502
[4] Iorio, R. J., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations, t. 11, 1031-1081, (1986) · Zbl 0608.35030
[5] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., t. 58, 181-205, (1975) · Zbl 0343.35056
[6] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Appl. Math., Advances in Math. Suppl. Studies, t. 8, 93-128, (1983), Academic Press
[7] Nagumo, M., Über das anfangswertproblem partieller differentialgleichungen, Japan. J. Math., t. 18, 41-47, (1942) · Zbl 0061.21107
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