Global well-posedness for a class of hyperbolic equation with nonlinear weak damping term and Hartree type nonlinearity

In this paper, we consider the global well-posedness of the initial boundary value problem for a class of hyperbolic equation featuring nonlinear weak damping term and Hartree type nonlinearity. We begin by proving the existence and uniqueness of the local solution. A key challenge lies in analyzing the interaction and competition between nonlinear dissipation and the Hartree type nonlinearity with nonlocal effects. We develop the corresponding framework of the potential well theory, which allows us to establish the dependence of the dynamical behaviors of the solution on initial data. Specifically, we demonstrate results on the global existence and finite time blowup of the solution for subcritical and critical initial energy levels, as well as finite time blowup of the solution for non-positive and arbitrarily positive initial energy levels. Furthermore, we explore the long-term dynamics and continuous dependence of the global solution on the initial data and the damping coefficient, and estimate the blowup time of the blowup solution.