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On the Cauchy problem for a nonlocal nonlinear Schrödinger equation occurring in plasma physics. (English) Zbl 0780.35104
Summary: The goal is to study the Cauchy problem for some generalization of the following system: \[ i\varphi_ t+ \Delta\varphi=-\text{div} (|\nabla\psi |^ 2 \nabla \psi), \qquad\Delta\psi=\varphi. \] In particular, we construct solutions in the energy space associated to this system. We give some sufficient conditions on the initial data which ensure that the solutions are global, but we show that in some cases, finite time blow up occurs.

35Q55 NLS equations (nonlinear Schrödinger equations)
35G25 Initial value problems for nonlinear higher-order PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow