Advances in Differential Equations

On a nonlinear dispersive equation with time-dependent coefficients

Corinne Laurey

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Abstract

As a first step, we consider an evolution linear problem, the symbol of which is a real polynomial of degree three with time-dependent coefficients. We get for this problem smoothing effects known when these coefficients are constant. In particular, by using the theory of Calderón-Zygmund operators and the David and Journé T1 Theorem, we establish a local smoothing effect on the solution of the linear problem. In a second step, we study a nonlinear dispersive equation the linear part of which is the one studied above. We use the previous smoothing properties and a regularization method to establish that the Cauchy problem is locally well-posed in the Sobolev spaces $H^s(\mathbb R)$ for $s>3/4$.

Article information

Source
Adv. Differential Equations Volume 6, Number 5 (2001), 577-612.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1357141856

Mathematical Reviews number (MathSciNet)
MR1826722

Zentralblatt MATH identifier
1003.35118

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B35: Stability 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q60: PDEs in connection with optics and electromagnetic theory

Citation

Laurey, Corinne. On a nonlinear dispersive equation with time-dependent coefficients. Adv. Differential Equations 6 (2001), no. 5, 577--612. https://projecteuclid.org/euclid.ade/1357141856.


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