Blow-up of radially symmetric solutions for a cubic NLS type system in dimension 4
Maicon Hespanha, Ademir Pastor
TL;DR
The paper studies an energy-critical cubic NLS-type system in $\mathbb{R}^4$ modeling the interaction of an optical beam with its third harmonic. It obtains the existence of ground-state solutions via concentration-compactness and identifies a sharp vector Sobolev-type inequality, linking ground-state energy to optimal constants. The main dynamical result shows that radially symmetric solutions with energy below the ground-state energy but kinetic energy above the ground-state blow up in finite time, using a convexity/virial framework adapted to the vector setting. An appendix extends the blow-up result to the resonance case without radial symmetry, highlighting the robustness of the blow-up mechanism in this coupled system.
Abstract
This paper is concerned with a cubic nonlinear Schrödinger system modeling the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We are mainly interested in the so-called energy-critical case, that is, in dimension four. Our main result states that radially symmetric solutions with initial energy below that of the ground states but with kinetic energy above that of the ground states must blow-up in finite time. The proof of this result is based on the convexity method. As an independent interest we also establish the existence of ground state solutions, that is, solutions that minimize some action functional. In order to obtain our existence results we use the concentration-compactness method combined with variational arguments. As a byproduct, we also obtain the best constant in a vector critical Sobolev-type inequality.