Singularities and Global Solutions in the Schrodinger-Hartree Equation

In 1922, Louis de Broglie proposed wave-particle duality and introduced the idea of matter waves. In 1925, Erwin Schrodinger, proposed a wave equation for de Broglie’s matter waves. The Schrodinger equation is described using the de Broglie’s matter wave, which takes the wave function, and describes its quantum state over time. Herein, we study the generalized Hartree (gHartree) equation, which is a nonlinear Schrodinger type equation except now the nonlinearities are a nonlocal (convolution) type. In the gHartree equation, the influence on the behavior of the solutions is global as opposed to the case of local (power type) nonlinearities. Our first goal is to understand the behavior of finite energy solutions. We start with proving the local existence and then extend to the global existence for small data. We then, in the energy-subcritical critical regime, classify the behavior of finite energy solutions under the mass-energy assumption identifying the sharp threshold (depending on the size of the initial mass and gradient) for global (scattering) versus finite time (blow-up) solutions. Next, we revisit the problem of scattering and give an alternative proof of scattering, for both NLS and gHartree equations in the radial setting. The alternative approach provides a simpler proof of scattering, which might also be useful for other contexts. Our next aim is to understand the phenomenon of wave collapse (blow-up, the sudden energy transfer from higher levels to lower ones), i.e., solutions with finite time of existence. We first give a sufficient condition for finite time blow-up for the large data and give examples of the various thresholds available in a variety of cases (energy-subcritical, critical and supercritical) for Gaussian data. We then investigate stable singularity formations in the mass-critical gHartree equation, and in particular, rigorously prove a stable blow-up formation in dimension 3. We observe that the nonlocal nonlinearity does not destroy the blow-up dynamics, similar to the local nonlinearities. On the other hand, one of the necessary properties, namely the spectral property required for the blow-up analysis, is modified remarkably. Nevertheless, we are able to prove that stable blow-up occurs with a self-similar profile at the square root rate with a logarithmic correction. Finally, we present the reader with the conclusion and possible future research directions, wrapping up the dissertation