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Wellposedness for Zakharov systems with generalized nonlinearity. (English) Zbl 0921.35162
The paper is devoted to the asymptotic behaviour of solutions of the Cauchy problem \[ i\partial u/\partial t+\Delta u=nf\bigl(| u|^2 \bigr)u,\;\square n=\Delta \biggl(F\bigl(| u|^2\bigr)\biggr) \] \[ (u,n, \partial n/\partial t)(0)= (\varphi,a,b) \] where \(F(s)=s^m\), \(f(s)=ms^{m-1}\) \((m \geq 2\), \(u:\mathbb{R}^d \times\mathbb{R} \to\mathbb{C},n: \mathbb{R}^d\times \mathbb{R}\to \mathbb{R})\). Certain modification of the methods by J. Bourgain and J. Colliander [Int. Math. Res. Not. 1996, No. 11, 515-546 (1996; Zbl 0909.35125)] are introduced to establish the following result: The problem is locally well-posed in \(H^s \times L^2\times \widehat H^{-1}\) with \(d=1,2\) and \(s<1\). If moreover \((\varphi, a,b) \in\cap_{\sigma <s}H^\sigma \times H^{\sigma-1} \times\widehat H^{\sigma-2}\) where \(s\geq 1\), then the solution \((u(t),n(t),n(t))\) belongs to the same space for fixed \(t\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
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