Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

Boundary Value Problems

Yongbin Wang¹  Binhua Feng² 

[1] Department of Basic Teaching and Research, Qinghai University;Department of Mathematics, Northwest Normal University;

关键词: Nonlinear Schrödinger–Choquard equation;    Sharp thresholds;    Blow-up;   

DOI  :  10.1186/s13661-019-01310-6

来源: DOAJ

【 摘 要 】

Abstract In this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation i ψ t + Δ ψ = λ 1 | ψ | p 1 ψ + λ 2 ( I α ∗ | ψ | p 2 ) | ψ | p 2 − 2 ψ . $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$ We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the L 2 $L^{2}$ -critical case. Our obtained results extend and improve some recent results.

【 授权许可】

Unknown