Kato, Tosio; Masuda, Kyûya Nonlinear evolution equations and analyticity. I. (English) Zbl 0622.35066 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 455-467 (1986). 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 Cited in 1 ReviewCited in 99 Documents MSC: 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 Keywords:nonlinear evolution equation; Banach space; Lyapunov functions; Korteweg- de Vries; Benjamin-Ono equations; analytic continuation × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Bourbaki, N., Fonctions d’une variable réelle (1949), Hermann: 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 · Zbl 0549.34001 [7] Nagumo, M., Über das Anfangswertproblem partieller Differentialgleichungen, Japan. J. Math., t. 18, 41-47 (1942) · Zbl 0061.21107 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.