Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

Abstract

We consider the quadratic nonlinear Schrödinger systemwhere 1 ≤ d ≤ 6 and κ > 0. In the lower dimensional case d = 1, 2, 3, it is known that the H 1 -solution is global in time. On the other hand, there are finite time blow-up solutions when d = 4, 5, 6 and κ = 1/2. The condition of κ = 1/2 is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when d = 5, 6 and κ = 1/2 and we show blow-up or grow-up when d = 4.

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

“…In [18] the authors, among other things, proved the existence of ground state solutions for (1.4), when κ > 0, and used these solutions to give a sharp criterion for the existence of global H 1 solutions in the mass-resonance case and n = 4. This kind of result was extended to the non-mass-resonance case (κ = 1 2 ) in [23], where the authors showed a blow-up result when the initial data is radial in dimensions n = 5 and n = 6. Some very recent works without mass-resonance condition have been appeared.…”

“…From the mathematical point of view the interest in nonlinear Schrödinger systems with quadratic interactions has been increased in the past few years (see [7], [8], [15], [17], [18], [23], [24], [25], [33], [34], [36], [39] and references therein). So, in [34] we initiated the study of system (1.1) with general quadratic-type nonlinearities.…”

Scattering for the quadratic nonlinear Schrödinger system in $\mathbb{R}^5$ without mass-resonance condition

“…Recently, the second author, Kishimoto, and the third author [11] obtained the scattering below the ground state in the L 2 -critical case, i.e., d = 4, under the assumption of radial symmetry. Moreover, they also showed the finite time blow-up result below the ground state under the assumption of radial symmetry in [12].…”

“…Remark 1.1. In the opposite case , i.e., K(u 0 , v 0 ) < 0, the second author, Kishimoto, and the third author [12] show that the finite time blow-up occurs in both time directions. And so, the behavior of the radially symmetric solution to (NLS) below the ground state completely determined by the sign of the functional K at initial time.…”

Global dynamics below the ground state for the focusing Schrödinger equation with a potential

“…The proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du-Wu-Zhang in [6]. We can exclude the possibility of the growing-up result by the argument in [23], [15], and [10] if "the data and the potential are radially symmetric" or "the data has finite variance".…”