Schrödinger-Poisson方程爆破解的动力学性质

Dynamical Properties of Blow-Up Solutions for the Schrödinger-Poisson Equation

摘要: 本文研究质量临界Schrödinger-Poisson方程的爆破解动力学性质。具体地，利用最佳Gagliardo-Nirenberg不等式及经典非线性薛定谔方程基态的变分刻画，在解$u\left(t\right)$ 于$T^{*}$ 时刻发生爆破的假设下，证明了当$t\to T^{*}$ 时$u\left(t\right)$ 不存在$L^2$ -强极限。进一步，当初值$u_0$ 具有径向对称性时，得到了爆破解$u\left(t\right)$ 会在原点处出现$L^2$ -集中现象。

Abstract: This paper investigates the dynamical properties of blow-up solutions to the mass-critical Schrödinger-Poisson equation. By using the sharp Gagliardo-Nirenberg inequality and the variational characterization of the ground state for the classical nonlinear Schrödinger equation, we prove that if a solution $u\left(t\right)$ blows up at a finite time $T^{*}$ , then $u\left(t\right)$ admits no strong limit in $L^2$ as $t\to T^{*}$ . Furthermore, when the initial datum $u_0$ is radially symmetric, the blow-up solution $u\left(t\right)$ exhibits $L^2$ -concentration at the origin.