Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896793 | Journal of Functional Analysis | 2018 | 31 Pages |
Abstract
We consider the nonlinear Schrödinger equationiut+Îu=λ|u|2Nu in all dimensions Nâ¥1, where λâC and âλâ¤0. We construct a class of initial values for which the corresponding solution is global and decays as tââ, like tâN2 if âλ=0 and like (tlogâ¡t)âN2 if âλ<0. Moreover, we give an asymptotic expansion of those solutions as tââ. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at u=0. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Thierry Cazenave, Ivan Naumkin,