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Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system

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  • In this paper, we consider the following equivariant defocusing Chern-Simons-Schrödinger system,

    $\begin{eqnarray}i\partial_{t}\phi+\Delta\phi = \frac{2m}{r^2}A_{\theta}\phi+A_{0}\phi+\frac{1}{r^2}A_{\theta}^2\phi-\lambda|\phi|^{p-2}\phi,\\ \partial_rA_{0} = \frac{1}{r}(m+A_{\theta})|\phi|^2,\\ \partial_tA_{\theta} = rIm(\bar{\phi}\partial_{r}\phi),\\ \partial_rA_{\theta} = -\frac{1}{2}|\phi|^2r,\\ A_r = 0.\end{eqnarray}$

    where $ \phi(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $ is a complex scalar field, $ A_\mu(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $ is the gauge field for $ \mu = 0, 1, 2 $, $ A_r = \frac{x_1}{|x|}A_1+\frac{x_2}{|x|}A_2 $, $ A_{\theta} = -x_2A_1+x_1A_2 $, $ \lambda<0 $ and $ p>4 $.

    When $ p>4 $, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.

    Mathematics Subject Classification: 35A01, 35Q41, 35Q55.


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