##
**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.

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.

Reviewer: Giulio Soliani (Lecce)

### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35B44 | Blow-up in context of PDEs |

35C08 | Soliton solutions |