On the solvability of a mixed problem for a nonlinear equation of Schrödinger type. (English. Russian original) Zbl 0585.35019

Vladimirov, M. V.

Sov. Math., Dokl. 29, 281-284 (1984); translation from Dokl. Akad. Nauk SSSR 275, 780-783 (1984).

For a bounded domain \(G\subset {\mathbb{R}}^ n\) an initial-boundary-value problem for a nonlinear Schrödinger equation is considered: \[ \partial u/\partial t+i\Delta u+i\alpha | u|^ pu+\beta | u|^ qu=0,\quad u|_{\partial G\times [0,T]}=0,\quad u(x,0)=u_ 0(x). \] It is shown that for \(0\leq p<q\), \(\beta >0\) and any real \(\alpha\) a global (weak) solution exists. For \(p=2k\), \(k\in {\mathbb{N}}\), \(q\geq 0\) and \(\beta\geq 0\) a uniqueness result is given.

Reviewer: N.Jacob

MSC:

35J10	Schrödinger operator, Schrödinger equation
35J65	Nonlinear boundary value problems for linear elliptic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
49M15	Newton-type methods

Keywords:

existence of solution; initial-boundary-value problem; nonlinear Schrödinger equation; global; solution; uniqueness