On the Global Solutions of Abstract Wave Equations with High Energies

An important issue in the dynamics of an evolution equation is to characterize the initial data set that generates global solutions. This is an open problem for nonlinear partial differential equations of second order in time, with a nonlinear source term, and an arbitrary positive value of the initial energy. Recently, a new functional, $K$, has been proposed to achieve this goal, showing that its sign is preserved along the solutions, if some hypotheses on the initial data are satisfied. Trying to improve these results, the author realized that these hypotheses are satisfied only by the empty set. Here we prove this statement, and investigate another set of hypotheses, as well as the feasibility of preserving the sign of $K$ along the solutions. To analyze a broad set of evolution equations, we consider a nonlinear abstract wave equation.