In this paper, being investigated an initial - boundary value problem
for a one - dimensional wave equation with a nonlinear source of
variable order and nonlinear dissipation at the boundary. The existence
of a local solution of the problem under consideration is proved. Then
the question of the absence of global solutions is investigated.
Depending on the relationship between the order of growth of the
nonlinear source and the nonlinear boundary dissipation, different
results are obtained on the blow - up of weak solutions in a finite time
interval.