Blow-up dynamics in the mass super-critical NLS equations

Stable blow‐up dynamics in the ‐critical and ‐supercritical generalized Hartree equation

“…From the theory of Volterra integral equation (see exact statements in Ref. 23, as well as the application in the NLS case, which is following (Ref. 49 We next note that is the solution not only to Equation (26) or (27), but also to the differential equation (16), and thus, differentiating twice classically, it gives ∈ ℂ 2 [0, ∞), finishing the proof.…”

“…In Ref. 23, we obtained solutions of this NLS system (101), hence, we can use them as our initial guess. Similar to the NLS case, there are multiple solutions to the system (100), and we are able to find some of them, though not all these solutions are profiles for stable blow-up.…”

“…We choose the corresponding multi-bump solutions from NLS equation in Ref. 23, as the initial guess, which is suitable in the gHartree setting.…”

“…48 and Ref. 23 for the NLS). For that, from (), we claim that if is bounded, so is .…”

“…20, 21, also Refs. 22, 23, see Section 3 for details). In particular, we will investigate the blow‐up rate and blow‐up profiles of singular solutions to the gHartree equation () in the critical and supercritical settings.…”