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On a nonlinear dispersive equation with time-dependent coefficients. (English) Zbl 1003.35118

Summary: As a first step, we consider a linear evolution 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 Calderon-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 \[ iu_t+ia_3 u_{xxx}+ a_2 (t)u_{xx}= u\bigl(g*|u|^2 \bigr)+i{\partial \over\partial x}\biggl[u \bigl(g* |u|^2 \bigr)\biggr], \] 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 \(s>3/4\).

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35Q55 NLS equations (nonlinear Schrödinger equations)
35B65 Smoothness and regularity of solutions to PDEs
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