The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^ s\). (English) Zbl 0706.35127

Consider the Cauchy problem for the nonlinear Schrödinger equation in \({\mathbb{R}}^ N\), \[ (NLS)\quad iu_ t+\Delta u+\lambda | u|^{\alpha}u=0,\quad u(0)=\phi. \] Here, \(\phi\) is a given initial value, \(\lambda\) is a real parameter and \(\alpha >0\) is arbitrary (up to technical conditions that intervene only for dimensions \(N\geq 7)\). It is established that if \(0\leq s<N/2\) and \(\alpha\leq 4/(N-2s)\), then problem (NLS) is locally well posed in the Sobolev space \(H^ s({\mathbb{R}}^ N)\). In the case \(\alpha =4/(N-2s)\), if \(\| (-\Delta)^{s/2}\phi \|_{L^ 2}\) is sufficiently small, then the solution is global and decays as \(t\to \infty\) at the same rate as the free solution. The method relies on estimates for \(e^{it\Delta}\) in certain Besov spaces, on estimates of \(| u|^{\alpha}u\) in the same Besov spaces, and on a fixed point argument.
Reviewer: Th.Cazenave


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