Some remarks on the nonlinear Schrödinger equation with fractional dissipation

Abstract

Abstract. We consider the Cauchy problem for the L 2 -critical focussing nonlinear Schrödinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for ∇u(t) L 2 .

On the Cauchy problem for the nonlinear Schrodinger equations including fractional dissipation with variable coefficient

“…Note that our proof is very close to the case of a ( t )= a >0(see Darwich and Molinet). Actually, as noticed in Planchon and Raphaël, we only need to prove that in the log‐log regime, the L 2 norm does not grow, and the growth of the energy (resp the momentum) is below (resp ).…”

“…Proof See Darwich and Molinet ,. Lemma 4.1 In the same way as in Darwich and Molinet, we have the following lemma: □…”

“…An important feature of this estimate of H 1 flavor is that it relies on a flux computation in L 2 . This allows one to recover the asymptotic laws for the geometrical parameters (24) and to close the bootstrap estimates of the log-log regime.…”

“…Darwich and Molinet have studied the global existence and the blow‐up solutions of in the case where a ( t )= a >0 and s >0.…”

A Note on a Damped Focusing Inhomogeneous Choquard Equation

“…In [18], M. Ohta and G. Todorova established that the Cauchy problem associated to (1.2) is well posed in the energy space and the solution is global for large damping. For other modifications of the classical equation (1.2), see also [6,7,24,25]. It is thus quite natural to complete the nonlinear Choquard equation by a linear dissipative term to take into account some dissipation phenomena.…”