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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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