×

Time decay of solutions to the Schrödinger equation in exterior domains. I, II. (English) Zbl 0684.35019

The time evolution given by the free particle Schrödinger operator on the complement, D, of a star-shaped bounded domain in \({\mathbb{R}}^ n\) is studied. Dirichlet boundary conditions are considered. Upper bounds for the decay rates of the \(L^ p(D)\)-norm of the evolved state, u(t), are obtained. The bounds obtained in the first part are improved in the second one. For orientation let us reproduce one particular result: for \(n\geq 5\), \(2\leq p\leq 2n/(n-4)\) \[ \| u(t)\|_ p\leq const. I^{1/2}(1+t)^{-n(1-2/p)/2}, \] where I is a functional of the initial value, u(0).
Reviewer: G.Nenciu

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35K10 Second-order parabolic equations
PDF BibTeX XML Cite
Full Text: Numdam Numdam EuDML

References:

[1] A. Friedman , Partial Differential Equations: Holt , Rinehart and Winston , New York , 1969 . MR 445088 | Zbl 0224.35002 · Zbl 0224.35002
[2] Y. Jing-Qi , Comportement à l’infini des solutions d’une equation de Schrödinger non linéaire dans un domaine extérieur , C. R. Acad. Sc. Paris , t. 294 , 1982 , p. 163 - 166 . MR 651435 | Zbl 0511.35078 · Zbl 0511.35078
[3] O.A. Ladyzhenskaya , The Mathematical Theory of Viscous Incompressible Flow , 2 nd English ed., Gordon and Breach , New York , 1969 . MR 254401 | Zbl 0184.52603 · Zbl 0184.52603
[4] O.A. Ladyzhenskaya and N.N. Ural’ceva , Linear and Quasilinear Equations of Elliptic Type , Academic Press , New York , 1968 . · Zbl 0164.13002
[5] Y. Tsutsumi , Global solution of the nonlinear Schrödinger equation in exterior domains , Comm. PDE , t. 8 , 1983 , p. 1337 - 1374 . MR 711442 | Zbl 0542.35027 · Zbl 0542.35027
[6] K. Yosida , Functional Analysis , Springer-Verlag , Berlin , Heidelberg , New York , 1971 . Zbl 0217.16001 · Zbl 0217.16001
[7] Y. Chen , Global existence of the solution of nonlinear Schrödinger equations in exterior domains , Acta Mathematicae Applicatae Sinica , t. 2 , ( 3 ), 1985 , p. 191 - 212 . Zbl 0641.35011 · Zbl 0641.35011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.