Blow-up results of virial type for Zakharov equations
The author considers the collapse (blow-up) problem, i.e., appearance of a singularity in the solution to the Cauchy problem, for the multidimensional Zakharov's system: \[ iu_t+\Delta u-nu=0, \qquad c_0^{-2}n_{tt}= \Delta(n+|u|^2), \] where \(u\) and \(n\) are complex and real variables, and \(\Delta\) stands for the Laplacian. Numerical simulations of this system strongly suggest that a singularity develops in a finite time from a generic initial condition. However, the classical virial identities, that help to rigorously prove existence of the collapse in the nonlinear Schrödinger equation, which is the limit case of this system when \(c^2_0\to\infty\), does not allow to obtain the proof for the present system, because the corresponding virial quantity lacks the necessary positive definiteness. Therefore, instead of treating the general case, the author concentrates on the particular case of the radially symmetric initial condition. In this special case, the virial identity, together with the Hamiltonian representation of the system, allows him to produce a proof of existence of the singularity in the generic solution. However, although it seems very feasible that the singularity will always develop in a finite time, the available proof formally admits a possibility that the singularity may also set in at \(t\to\infty\).
scientific article; zbMATH DE number 878044