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Quelques propriétés qualitatives de l'équation de Schrödinger non-linéaire

Abstract : The work presented in this thesis deals about the Schrödinger equation with single power of nonlinearity. In a first part, we study global solutions which have a scattering state in a weighted Sobolev space. Since the free operator of Schrödinger is not an isometry on this space, we would like to know if such solutions converge to their scattering state. The converse is also studied. In a second part, we show that the solutions decay at most as those of the linear Schrödinger equation. A third part gives necessary conditions and sufficient conditions to obtain global solutions in the supercritical case. In a fourth part, we simplify the result of Kenji Nakanishi. He shows in the dissipative case that under suitable conditions on the nonlinearity, each solution with finite energy has a scattering state. In this part, we give a proof which does not require the Besov spaces, unlike the original proof, since the result occurs in the energy space. A last part deals about the regularity of some self-similar solutions.
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Contributor : Pascal Bégout <>
Submitted on : Monday, February 21, 2005 - 6:06:56 PM
Last modification on : Wednesday, December 9, 2020 - 3:06:03 PM
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  • HAL Id : tel-00007378, version 2


Pascal Bégout. Quelques propriétés qualitatives de l'équation de Schrödinger non-linéaire. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2001. Français. ⟨tel-00007378v2⟩



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