Exponential stability of solutions to the Schrödinger–Poisson equation

Joackim Bernier; Nicolas Camps; Benoît Grébert; Zhiqiang Wang

doi:10.3934/dcds.2024064

Article Contents

2024, Volume 44, Issue 11: 3398-3442. Doi: 10.3934/dcds.2024064

This issue Previous Article Exceptional sets for geodesic flows of noncompact manifolds Next Article Uniqueness and nondegeneracy of ground states for the Schrödinger-Newton equation with power nonlinearity

Exponential stability of solutions to the Schrödinger–Poisson equation

1.
Laboratoire de Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 44322 Nantes Cedex 03, France
2.
Chern Institute of Mathematics, Nankai University, Tianjin 300071, China

^*Corresponding author: Zhiqiang Wang
^*Corresponding author: Zhiqiang Wang

Received: January 1, 2024

Revised: April 1, 2024

Early access: May 13, 2024

Published online: May 13, 2024

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Abstract

We prove an exponential stability result for the small solutions of the Schrödinger–Poisson equation on the circle without exterior parameters in Gevrey class. More precisely we prove that for most of the initial data of Gevrey-norm smaller than $ \varepsilon $ small enough, the solution of the Schrödinger–Poisson equation remains smaller than $ 2\varepsilon $ for time of order $ \exp \big(\alpha \frac{|\log \varepsilon|^2}{\log|\log \varepsilon|} \big) $. We stress out that this is the optimal time expected for PDEs as conjectured by Jean Bourgain in [18].

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Mathematics Subject Classification: Primary: 37K45, 37K55; Secondary: 35Q55.

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References

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