Dynamical Properties Of Blow-up Solutions For Nonlinear Schrodinger Equations With Potentials

In this paper, we make investigations on dynamical properties of the blow-up solutions for nonlinear Schrodinger equations with potentials. A sharp sufficient condition for blow-up solutions is obtained. Furthermore, an estimate on upper and lower bound of the blow-up rate as well as L2-mass concentration properties of blow-up solutions are established.Firstly, we consider the nonlinear Schrodinger equationswhere p is a parameter, k(x) is a bounded and differentiate function on Rn. For the equation with critical power and supercritical power, we study the relationship between blow-up solutions and the corresponding energy , and prove the existence of blow-up solutions with a developed energy method. It generalizes the results in [11]. Furthermore, by exploiting the variational characterization of the ground state solution for the corresponding scalar field equation, we obtain a sharp sufficient condition of blow-up solutions for the following nonlinear Schrodinger equation with a harmonic potentialSecondly, some discussions are given to the following nonlinear Schrodinger equation with a harmonic potential, which is well known as a model for describing the Bose-Einstein condensate with attractive inter-particle interactionsand the nonlinear Schrodinger equation with a Stark potentialwhere ω>0, k > 0, a ∈ Rn\ {0}. Making full use of the results of the classical Schrodinger equation and the spectral property, we obtain an estimate on upper and lower bound of the blow-up rate for both equation (0.3) and equation (0.4).Lastly, as a more elaborate description of the local behavior for blow-up solutions, we study an L2-mass concentration phenomenon of blow-up solution which occurs at blow-up time. For n = 2(or R2), we prove that the blow-up solutions of equation (0.3) have the L2-mass concentration properties, by using a scaling argument and a compactness lemma. For n > 2(or Rn), we prove that L2-mass concentration phenomena of the blow-up solutions only occur under some conditions. In particular, when the initial data has a spherical symmetry, an L2-mass concentration phenomenon occurs at the origin O.