Abstract:
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.
Keywords:abstract wave equation, global solutions, high energies.
