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
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