The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases

This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross–Pitaevskii equation describing the Bose–Einstein condensation of trapped dipolar quantum gases. Via a construction of new cross-constrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum.