Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

Abstract

Abstract. In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than (T − t) −1 , the expected one. Moreover, we show that blowup cannot occur on the boundary, under certain geometric conditions on the domain.

Existence and Stability of the log–log Blow-up Dynamics for the L 2-Critical Nonlinear Schrödinger Equation in a Domain

“…Note that such a result is known to be true when Ω = R N ; its proof relies heavily on the pseudo-conformal symmetry (5), which no longer holds in a domain. Some preliminary informations on the blow up speed for the critical mass finite time blow-up solutions have been obtained by Banica, [2].…”

“…We refer to [24] for more detailed computations. Let us recall thatQ satisfies (2) , Ψ (1) given by (49) and:…”

Classification of minimal mass blow-up solutions for an $${L^{2}}$$ L 2 critical inhomogeneous NLS

“…The following observation of Banica [1] is relevant in this context. For u ∈ H 1 (R N ), θ ∈ C ∞ 0 (R N ) real-valued and s ∈ R, we have ∇(ue isθ ) = (∇u + isu∇θ)e isθ , and so By integrating with respect to x ∈ R N , we get…”

“…In fact, Theorem 2.5 of [7] extends a result of Weinstein [14] from the case b = 0 to the case 0 < b < min{2, N }, and Theorem 1 above extends the classification result of Merle [10] to the case 0 < b < min{2, N }. A comprehensive review of the theory of blow-up solutions for the classic focusing NLS can be found in [12], where a proof of Merle's result [12,Theorem 4.1] is presented, which is based on more recent arguments-notably a refined Cauchy-Schwarz inequality due to Banica [1].…”

Threshold solutions for the focusing 3D cubic Schrödinger equation

“…Appendix contains the proof of coercivity of the quadratic form introduced in Section 3 where we follow Weinstein [33] and a useful inequality, the original idea of which is due to Banica [2].…”