The Blow-up Dynamics Of Nonlinear Schr(?)dinger Equation With A Repulsive Harmonic Potential

Abstract: In this paper, we study the dynamical properties of the blow-up so-lutions for nonlinear Schr(o|¨)dinger equation with a repulsive harmonic potential.An estimate on the upper and lower bounds of the blow-up rate of blow-upsolution with small super-critical mass is obtained. Blow-up profiles for theradially blow-up solutions, L~2-mass concentration and rate of L~2-mass concen-tration are obtained. In particular, using the concentration compact principle,upper bound of L~2-weak limitation for blow-up solutions and the limit behaviorof minimal blow-up solutions are obtained.Consider the following nonlinear Schro¨dinger equation whereωis a positive parameter; is the Laplace operatoron R~N; u = u(t,x): [0,T)×R~N→C is the complex valued wave function,0 < T≤∞; N is the space dimension.At first, using Merle and Rapha¨el's arguments as well as the transformationproposed by Carles, blow-up rate of blow-up solutions for the Cauchy problem(0-1)-(0-2)is obtained, as follows.where C1 and C2 are two positive constants. Hence, the results in [10] aregeneralized.Secondly, Zhang's[65] showed a result which enlightens a rather surprisinganalysis between the study of nonlinear Schr(o|¨)dinger equations with and withouta harmonic potential. We choose the characterization of classical nonlinearSchr(o|¨)dinger equation without any potential to describe the blow-up solutionsof Cauchy problem (0-1)-(0-2). In terms of the methods used in the classicalnonlinear Schr(o|¨)dinger equation, blow-up profiles, L2-mass concentration andrate of L2-mass concentration for the radially blow-up solutions of the Cauchyproblem (0-1)-(0-2) are obtained. Hence, the results in [29, 55] are generalized. Thirdly, using the concentration compact principle proposed by Lions, weobtained an important estimate. Thus using the crucial estimate, we extendedthe L~2-mass concentration results to nonradially symmetric solutions. In par-ticular, the upper bound of L~2 weak-limitation of the blow-up solutions for theCauchy problem (0-1)-(0-2) is obtained by the crucial estimate.Lastly, in terms of Weinstein[59] and Zhang's [65] arguments, we choose theground state of classical nonlinear Schro¨dinger equation without any potentialto describe the minimal blow-up solutions of Cauchy problem (0-1)-(0-2). Usingthe scaling, concentration compact principle and conservation laws, the limitbehavior of the minimal blow-up solutions for the Cauchy problem (0-1)-(0-2)is obtained.