Blow-up results for systems of nonlinear Schrödinger equations with quadratic interaction

Abstract

We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.

On finite time blow-up for a 3D Davey-Stewartson system

“…where the real regular function η : R → R + ∪ {0} satisfies: supp η ⊂ (1, 2) and is normalized to one, namely R η(s)ds = 1. Observe that we have (1), while the local term (3.4) can be estimated as in Martel's paper [18] (see also [1,4,9] for similar results on different dispersive models). Precisely,…”

Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with $χ^3$ nonlinear response

“…To the best of our knowledge, the strategy of using an ODE argument -when classical virial estimates based on the second derivative in time of (localized) variance break down -goes back to the work [3], where fractional radial NLS is investigated. See instead [10,21] for some blow-up results for quadratic NLS systems.…”

“…Next we derive localized virial estimates for cylindrically symmetric solutions (we also mention here [2,10,19,20,24], for the qualitative analysis of dispersive-type equations in anisotropic spaces). To this end, we introduce…”

“…Thus the convexity argument is no-more applicable in our general setting. The proof of Theorem 1.2 above relies instead on an ODE argument, in the same spirit of our previous work [10], using localized virial estimates and the negativity property of the Pohozaev functional (see Lemma 5.1). We point-out that our result not only extends the one in [27] to radial and cylindrical solutions, but also extends it to the whole range of µ, γ > 0.…”

Ground state energy threshold and blow-up for NLS with competing nonlinearities

“…It is worth mentioning that the first early work for the classical focusing NLS equation in anisotropic spaces goes back to Martel [33]. See [2,3,15] for recent results for other classes of dispersive equations. From (5.3), and using again the identity (5.5), Note that in the last estimate, the contribution given by the lower order term |u| q+1 shall be discarded, as it accounts for a nonpositive contribution, hence (5.7) can be refined as (5.10) Indeed, by recalling that for a one-dimensional function f ∈ H 1 (R) .…”