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It is known that degenerate parabolic equations exhibit somehow different phenomena when we compare them with their elliptic counterparts. Thus, the problem of existence and properties of the Green function for degenerate parabolic boundary value problems is not completely solved, even after the contributions of [GN] and [GW4], in the sense that the existence problem is still open, even if the a priori estimates proved in [GN] will be crucial in our approach (...).
We study an Helium atom (composed of one nucleus and two electrons) submitted to a general time dependent electric field, modeled by the Hartree-Fock equation, whose solution is the wave function of the electrons, coupled with the classical Newtonian dynamics, for the position of the nucleus. We prove a result of existence and regularity for the Cauchy problem, where the main ingredients are a preliminary study of the regularity in a nonlinear Schrödinger equation with semi-group techniques and...
The theory of maximal monotone operators is applied to prove the existence of weak periodic solutions for a nonlinear nonlocal problem. The stability of these solutions is a consequence of the Lipschitz continuous assumption on the diffusivity matrix and the death rate.
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