Maximum Decay Rate for the Nonlinear Schrödinger Equation

Pascal Bégout (1, 2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LJLL - Laboratoire Jacques-Louis Lions

In this paper, we consider global solutions for the following nonlinear Schrödinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with $\lambda\in\R$ and $0\le\alpha<\frac{4}{N-2}$ $(0\le\alpha<\infty$ if $N=1).$ We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that $u(0)$ lies in the weighted Sobolev space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$ or in $L^2(\R^N),$ according to the different cases.

Domaines: Equations aux dérivées partielles [math.AP]

Pascal Bégout. Maximum Decay Rate for the Nonlinear Schrödinger Equation. Nonlinear Differential Equations and Applications, 2004, 11 (4), pp.451-467. ⟨10.1007/s00030-004-2003-7⟩. ⟨hal-00715811v2⟩