Blow-Up Solutions of Two-Coupled Nonlinear Schrödinger Equations in the Radial Case

Abstract

We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations

$$\left\{ {\matrix{{{\rm{i}}{u_t} + \Delta u + (|u{|^{2p}} + \beta |u{|^{p - 1}}|v{|^{p + 1}})u = 0,} \hfill \cr {{\rm{i}}{v_t} + \Delta v + (|v{|^{2p}} + \beta |v{|^{p - 1}}|u{|^{p + 1}})v = 0,} \hfill \cr {u(0,x) = {u_0}(x),\,\,\,\,\,\,v(0,x) = {v_0}(x),\,\,\,\,\,\,x \in {\mathbb{R}^N},\,\,t \ge 0.} \hfill \cr } } \right.$$

On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ((∣x∣u₀, ∣x∣v₀) ∈ L²(ℝ^N) × L²(ℝ^N)).