tailieunhanh - Đề tài "The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr¨odinger equation "

The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr¨dinger equation o By Frank Merle and Pierre Raphael Abstract We consider the critical nonlinear Schr¨dinger equation iut = −∆u−|u| N u o with initial condition u(0, x) = u0 in dimension N = 1. For u0 ∈ H 1 , local existence in the time of solutions on an interval [0, T ) is known, and there exist finite time blow-up solutions, that is, u0 such that limt↑ | Annals of Mathematics The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr odinger equation By Frank Merle and Pierre Raphael Annals of Mathematics 161 2005 157 222 The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrodinger equation By Frank Merle and Pierre Raphael Abstract We consider the critical nonlinear Schrodinger equation iut Au u Nu with initial condition u 0 x u0 in dimension N 1. For u0 G H1 local existence in the time of solutions on an interval 0 T is known and there exist finite time blow-up solutions that is u0 such that limt T - - lux t lL2 ro. This is the smallest power in the nonlinearity for which blow-up occurs and is critical in this sense. The question we address is to understand the blow-up dynamic. Even though there exists an explicit example of blow-up solution and a class of initial data known to lead to blow-up no general understanding of the blow-up dynamic is known. At first we propose in this paper a general setting to study and understand small in a certain sense blow-up solutions. Blow-up in finite time follows for the whole class of initial data in H 1 with strictly negative energy and one is able to prove a control from above of the blow-up rate below the one of the known explicit explosive solution which has strictly positive energy. Under some positivity condition on an explicit quadratic form the proof of these results adapts in dimension N 1. 1. Introduction . Setting of the problem. In this paper we consider the critical nonlinear Schroodinger equation iut Au u Nu t x G 0 T X R v u 0 x u0 x u0 RN C with u0 G H1 H1 RN in dimension N 1. The problem we address is the one of formation of singularities for solutions to 1 . Note that this equation is Hamiltonian and in this context few results are known. It is a special case of the following equation 2 iut Au u p-1u where 1 p N 2 and the initial condition u0 G H1. From a result of Ginibre and Velo 8 2 is locally .