Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations

Begout, Pascal ; Diaz, Jesus Ildefonso

Electronic Journal of Differential Equations, Tome 2014 (2014).

Résumé

Summary: "Sharp localized" solutions (i.e. with compact support for each given time t) of a singular nonlinear type Schrödinger equation in the whole space $\mathbb{R}^N$ are constructed here under the assumption that they have a self-similar structure. It requires the assumption that the external forcing term satisfies that $\mathbf{f}(t,x)=t^{-(\mathbf{p}-2)/2}\mathbf{F}(t^{-1/2}x)$ for some complex exponent $\mathbf{p}$ and for some profile function $\mathbf{F}$ which is assumed to be with compact support in $\mathbb{R}^N$. We show the existence of solutions of the form $\mathbf{u}(t,x)=t^{\mathbf{p}/2}\mathbf{U}(t^{-1/2}x)$, with a profile $\mathbf{U}$, which also has compact support in $\mathbb{R}^N$. The proof of the localization of the support of the profile $\mathbf{U}$ uses some suitable energy method applied to the stationary problem satisfied by $\mathbf{U}$ after some unknown transformation.

Détail

Classification : 35B99, 35A01, 35A02, 35B65, 35J60
Keywords: nonlinear self-similar Schrödinger equation, compact support, energy method