## On universality of blow up profile for $$L^2$$ critical nonlinear Schrödinger equation.(English)Zbl 1067.35110

The authors consider finite blow up solutions to the critical nonlinear Schrödinger (NLS) equation $iu_t= -\Delta u-|u|^{4/N}u\tag{1}$ with initial condition $$u_0\in H^1$$.
Although the existence of such solutions is known, the complete blow up dynamic is not understood so far. For a specific set of initial data, finite time blow up with a universal sharp upper bound on the blow up rate has been proved in earlier papers.
The authors establish the existence of a universal blow up profile which attracts blow up solutions in the vicinity of blow up time. This property relies on classification results of a new type for solutions to critical NLS equation.
In particular, a new characterization of soliton solutions is achieved, and a refined study of dispersive effects of NLS equation in $$L^2$$ will remove the possibility of self-similar blow up in energy space $$H^1$$. The authors continue the analysis of former papers. They investigate the question of the existence of a universal blow up profile at blow up time.
The analysis of this problem for Hamiltonian partial differential equations requires two types of information: i) rigidity properties of soliton solutions which are in the case treated by the authors the natural candidates of asymptotic profiles, and ii) dispersive results in the critical space $$L^2$$. Thus, the paper is constituted by two independent parts:
Part A, in which the authors establish various dynamical properties of negative energy solutions to (1) [where $$(t,x)\in [0,T)\times\mathbb{R}^N$$ and $$u(0,x)= u_0(x)$$, $$u_0:\mathbb R^N\to C$$, with $$u_0\in H^1= H^1(\mathbb R^N)$$ for $$N\geq 1$$ including global results in time $$t$$, and not only asymptotic estimates near blow up time.
Part B, in which the authors study $$L^2$$ dispersive properties of solutions to (1) which classify explicit the mass blow up solution $$S(t)$$ as the only non $$L^2$$ dispersive blow up solution.
The results listed in Part A and Part B allow the authors to prove the universal behaviour of negative energy solutions to (1) at blow up time.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B44 Blow-up in context of PDEs 35C08 Soliton solutions
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