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On the energy critical Schrödinger equation in 3D non-trapping domains. (English) Zbl 1200.35066
Summary: We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for \(H_0^1 (\Omega)\) data on any smooth, non-trapping domain \(\Omega \subset \mathbb R^3\). The key ingredient is a smoothing effect in \(L_x^5(L_t^2)\) for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35P25 Scattering theory for PDEs
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