We study stable blow‐up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger‐type equation with a nonlocal, convolution‐type nonlinearity:

First, we consider the ‐critical case in dimensions and obtain that a generic blow‐up has a self‐similar structure and exhibits not only the square root blowup rate , but also the log‐log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow‐up regime in the ‐critical nonlinear Schrödinger equation. In this setting, we also study blow‐up profiles and show that generic blow‐up solutions converge to the rescaled , a ground state solution of the elliptic equation .

中文翻译：

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临界和超临界广义Hartree方程中的爆破动力学稳定

我们在具有径向对称性的广义Hartree方程中研究稳定的爆炸动力学，该方程是具有非局部卷积型非线性的Schrödinger型方程：

首先，我们在尺寸上考虑临界情况，并得出一般爆炸具有自相似的结构，不仅表现出平方根爆炸率，而且表现出对数-对数校正（通过渐近分析和函数拟合），因此，其行为类似于临界非线性Schrödinger方程中的稳定爆破状态。在这种情况下，我们还研究了爆破曲线，并证明了一般的爆破解收敛于重新定标的椭圆方程的基态解。