BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE∗

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doi:10.1007/s10473-023-0423-x

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

{\begin{aligned} i u_{t} + Δ u + (| u |^{2 p} + β | u |^{p - 1} | v |^{p + 1}) u = 0, \\ i v_{t} + Δ v + (| v |^{2 p} + β | v |^{p - 1} | u |^{p + 1}) v = 0, \\ u (0, x) = u_{0} (x), v (0, x) = v_{0} (x), x \in R^{N}, t \geq 0. \end{aligned}

$\begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*}$ 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 | u_{0}, | x | v_{0}) \in L^{2} (R^{N}) \times L^{2} (R^{N})

$(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$ ).

Key words: Schrö, dinger equations, radial symmetry, blow-up, virial identity

Qianqian BAI, Xiaoguang LI, Li ZHANG. BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗[J].Acta mathematica scientia,Series B, 2023, 43(4): 1852-1864.

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