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Some remarks on the nonlinear Schrödinger equation in the critical case. (English) Zbl 0694.35170
Nonlinear semigroups, partial differential equations and attractors, Proc. Symp., Washington/DC 1987, Lect. Notes Math. 1394, 18-29 (1989).
[For the entire collection see Zbl 0673.00012.]
For the nonlinear Schrödinger equation \(iu+\Delta u=g(u)\), \(u(0,\cdot)=\phi (\cdot)\) in \({\mathbb{R}}^ n\), with \(g(u)=\lambda | u|^{\alpha}u\), it was proved, among other things, that when \(\alpha =4/n\), for every \(\phi \in L^ 2({\mathbb{R}}^ n)\), there exists a unique maximal solution \(u\in C([0,T^*);L^ 2({\mathbb{R}}^ n))\cap L^{\alpha +2}_{loc}([0,T^*);L^{\alpha +2}({\mathbb{R}}^ n))\). If \(n\geq 3\) and \(\alpha =4/(n-2)\), then, for every \(\phi \in H^ 1({\mathbb{R}}^ n)\), there exists a maximal solution \(u\in C([0,T^*);H^ 1({\mathbb{R}}^ n))\cap C^ 1([0,T^*);H^{-1}({\mathbb{R}}^ n))\). The cases discussed are critical. The approach is based on some sharp dispersive estimates for the linear Schrödinger equation.
Reviewer: J.Yong

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations