Blow‐up dynamics of L²−critical inhomogeneous fractional nonlinear Schrödinger equation

Abstract

The purpose of this work is to investigate the blow-up dynamics of L 2 −critical focusing inhomogeneous fractional nonlinear Schrödinger equation:For this, we establish a new compactness lemma related to the equation. By applying this lemma, we study the dynamical behavior for blow-up solutions for initial data ∈ H s (R N ) satisfying || || L 2 ≥ ||Q|| L 2 , where Q is the ground state solution of our problem. KEYWORDS blow up, compact lemma, dynamical behavior MSC CLASSIFICATION 35Q51; 35Q55

Local well-posedness of the fractional inhomogeneous NLS in Sobolev spaces

“…The first point (19) follows with Strichartz estimates and ( 25), (28). Moreover, the second point (20) follows similarly, indeed using the first point with the equality…”

“…Later on, for τ > 0, a dichotomy of global existence versus finite time blow-up of inter-critical H s radial solutions were derived in [20]. The instability of standing waves in the mass-critical and mass-super-critical regimes was obtained in [8,19]. The scattering of the inter-critical radial solutions was established by the third author [18] by use of Dodson-Murphy method [7].…”

L2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

“…In addition, we also investigated the decay of global solutions by Morawetz estimate for radial initial data. For the limiting profile of infinite time blow-up solutions at the blow-up time, in [25,26,[29][30][31][32], the authors investigated the L 2 concentration phenomenon for different models.…”

Local well‐posedness of a critical inhomogeneous Schrödinger equation

“…Later on, a dichotomy of global existence versus finite time blow-up of solutions under the ground state threshold was obtained in Peng and Zhao 9 following the spirit of Holmer and Roudenko. 10 Moreover, the dynamical behavior of the mass-critical nonglobal solution was considered in Peng et al 11 To the authors' knowledge, all the works dealing with the well-posedness of the above problem in Sobolev spaces treat the subcritical regime. It is the aim of this work to consider the Sobolev critical case.…”

“…Later on, a dichotomy of global existence versus finite time blow‐up of solutions under the ground state threshold was obtained in Peng and Zhao 9 following the spirit of Holmer and Roudenko 10 . Moreover, the dynamical behavior of the mass‐critical nonglobal solution was considered in Peng et al 11 …”