Finite time blow-up for the nonlinear Schr\xf6dinger equation in trapped dipolar quantum gases with arbitrarily positive initial energy

Abstract

In this paper, we study the blow-up criterion for the following nonlinear Schrödinger equation arising in trapped dipolar quantum gases: i∂tu=−12Δu+a2(x12+x22+x32)u+λ1|u|2u+λ2(K∗|u|2)u,(t,x)∈[0,T∗)×R3.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$i\\partial _{t}u=-\\frac{1}{2}\\Delta u+a^{2} \\bigl(x_{1}^{2}+x_{2}^{2}+x_{3}^{2} \\bigr)u+ \\lambda _{1} \\vert u \\vert ^{2}u+\\lambda _{2}\\bigl(K\\ast \\vert u \\vert ^{2}\\bigr)u,\\quad (t,x) \\in \\bigl[0,T^{*}\\bigr) \\times \\mathbb{R}^{3}. $$\\end{document} When a=0 or aneq 0, by constructing an invariant set, we establish a new blow-up criterion, which implies the existence of blow-up solutions with arbitrarily large initial energy. This result gives a positive answer to the problem left by Carles, Markowich, and Sparber (Nonlinearity 21:2569–2590, 2008).