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We study stable blow-up dynamics in the generalized Hartree equation withradial symmetry, a Schr\"odinger-type equation with a nonlocal,convolution-type nonlinearity: $iu_t+\Delta u +\left(|x|^{-(d-2)} \ast |u|^{p}\right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the$L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a genericblow-up has a self-similar structure and exhibits not only the square rootblowup rate but also the log-log correction (via asymptotic analysis andfunctional fitting). In this setting we also study blow-up profiles and showthat generic blow-up solutions converge to the rescaled $Q$, a ground statesolution of the elliptic equation $-\Delta Q+Q- \left(|x|^{-(d-2)} \ast |Q|^p\right) |Q|^{p-2} Q =0$.  We also consider the $L^2$-supercritical case in dimensions d=3,4. We derivethe profile equation for the self-similar blow-up and establish the existenceand local uniqueness of its solutions. As in the NLS $L^2$-supercriticalregime, the profile equation exhibits branches of non-oscillating, polynomiallydecaying (multi-bump) solutions. A numerical scheme of putting constraints intosolving the corresponding ODE is applied during the process of finding themulti-bump solutions. Direct numerical simulation of solutions to thegeneralized Hartree equation by the dynamic rescaling method indicates that the$Q_{1,0}$ is the profile for the stable blow-up. In this supercritical case, weobtain the blow-up rate without any correction. This blow-up happens at thefocusing level $10^{-5}$, and thus, numerically observable (unlike the$L^2$-critical case).  In summary, we find that the results are similar to the behavior of stableblowup dynamics in the corresponding NLS settings. Consequently, one may expectthat the form of the nonlinearity in the Schr\"odinger-type equations is notessential in stable blow-up.

Stable blow-up dynamics in the $L^2$-critical and $L^2$-supercritical generalized Hartree equation