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

Abstract: We study stable blow-up dynamics in the generalized Hartree equation with radial 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 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). In this setting we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled $Q$, a ground state solution 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 derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS $L^2$-supercritical regime, the profile equation exhibits branches of non-oscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ODE is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized 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, we obtain the blow-up rate without any correction. This blow-up happens at the focusing 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 stable blowup dynamics in the corresponding NLS settings. Consequently, one may expect that the form of the nonlinearity in the Schr\"odinger-type equations is not essential in stable blow-up.