  We study the stabilization and the wellposedness of solutions of the wave
equation with subcritical semilinearities and locally distributed nonlinear
dissipation. The novelty of this paper is that we deal with the difficulty that
the main equation does not have good nonlinear structure amenable to a direct
proof of a priori bounds and a desirable observability inequality. It is well
known that observability inequalities play a critical role in characterizing
the long time behaviour of solutions of evolution equations, which is the main
goal of this study. In order to address this, we truncate the nonlinearities,
and thereby construct approximate solutions for which it is possible to obtain
a priori bounds and prove the essential observability inequality. The treatment
of these approximate solutions is still a challenging task and requires the use
of Strichartz estimates and some microlocal analysis tools such as microlocal
defect measures. We include an appendix on the latter topic here to make the
article self contained and supplement details to proofs of some of the theorems
which can be found in the lecture notes of Burq and G\'erard (2001). Once we
establish essential observability properties for the approximate solutions, it
is not difficult to prove that the solution of the original problem also
possesses a similar feature via a delicate passage to limit. In the last part
of the paper, we establish various decay rate estimates for different growth
conditions on the nonlinear dissipative effect. We in particular generalize the
known results on the subject to a considerably larger class of dissipative
effects.
