Modified scattering for the critical nonlinear Schrödinger equation

Abstract

We consider the nonlinear Schrödinger equation $i{u}_t+\Delta u=\lambda |u{|}^{\frac{2}{N}}u$ in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{-\frac{N}{2}}$ if $\Im \lambda =0$ and like $(t\log t{)}^{-\frac{N}{2}}$ if $\Im \lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. 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.

References in corpus (2)

• Local existence, global existence, and scattering for the nonlinear Schrödinger equation
• A Fujita-type blowup result and low energy scattering for a nonlinear Schrö\-din\-ger equation

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