Well-posedness for degenerate Schrödinger equations

Massimo Cicognani; Michael Reissig

doi:10.3934/eect.2014.3.15

Article Contents

2014, Volume 3, Issue 1: 15-33. Doi: 10.3934/eect.2014.3.15

This issue Previous Article Existence and asymptotic behaviour for solutions of dynamical equilibrium systems Next Article Optimal control for stochastic heat equation with memory

Well-posedness for degenerate Schrödinger equations

Massimo Cicognani^1, and
Michael Reissig^2,

1.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna
2.
TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte Analysis, 09596 Freiberg

Received: March 2013

Revised: August 2013

Published: March 2014

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Abstract

We consider the initial value problem for Schrödinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.

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Mathematics Subject Classification: Primary: 35J10; Secondary: 35Q41.

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