On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data

Electronic Journal of Differential Equations

关键词: Mass-critical;    nonlinear Schrodinger equation;   

DOI  :  

来源: DOAJ

【 摘 要 】

For the focusing mass-critical nonlinear Schrodinger equation$iu_t+Delta u=-|u|^{4/d}u$, an important problem is toestablish Liouville type results for solutions with ground state mass.Here the ground state is the positive solution to elliptic equation$Delta Q-Q+Q^{1+frac 4d}=0$.Previous results in this direction were established in[12, 16, 17, 29] and they all require$u_0in H_x^1(mathbb{R}^d)$. In this paper, we consider the rigidity resultsfor rough initial data $u_0 in H_x^s(mathbb{R}^d)$ for any $s>0$.We show that in dimensions $dge 4$ and under the radial assumption,the only solution that does not scatter in both time directions(including the finite time blowup case) must be global and coincidewith the solitary wave $e^{it}Q$ up to symmetries of the equation.The proof relies on a non-uniform local iteration scheme, the refinedestimate involving the $P^{pm}$ operator and a new smoothing estimatefor spherically symmetric solutions.

【 授权许可】

Unknown