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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