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On the blow up phenomenon of the critical nonlinear Schrödinger equation. (English) Zbl 1099.35132

Consider the following problem \[ i\partial_t u + \Delta u = k | u| ^{4/d}u,\quad u(0,x) = u_0(x), \tag{1} \] where \(u_0\in L^2(R^d).\) It is known that in the sub-critical case (\(| u| ^{\alpha}u, 0<\alpha<4/d,\) instead of \(| u| ^{4/d}u)\) or in the case of sufficiently small \(u_0\), problem can be solved globally. However, for large data (in the critical case) blow up may occur. When the initial data \(u_0\) belongs to \(H^1\), we know that if \(\| u_0\| _{L^2}<\| Q\| _{L^2},\) where \(Q\) is the unique positive radial solution of the elliptic problem \[ \Delta Q - Q + | Q| ^{4/d}Q = 0, \] then the solution of (1) is global and the above bound is optimal.
The aim of this paper is to give a description of the blow up solution of (1) in one and two space dimensions. The author defines the minimal mass as the least \(L^2\)-norm needed to ignite the blow up phenomenon and he stresses the role of the solutions with minimal mass in the collapse mechanism.
Reviewer: Anh Cung (Hanoi)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
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