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We study the stabilization and the wellposedness of solutions of the waveequation with subcritical semilinearities and locally distributed nonlineardissipation. The novelty of this paper is that we deal with the difficulty thatthe main equation does not have good nonlinear structure amenable to a directproof of a priori bounds and a desirable observability inequality. It is wellknown that observability inequalities play a critical role in characterizingthe long time behaviour of solutions of evolution equations, which is the maingoal of this study. In order to address this, we truncate the nonlinearities,and thereby construct approximate solutions for which it is possible to obtaina priori bounds and prove the essential observability inequality. The treatmentof these approximate solutions is still a challenging task and requires the useof Strichartz estimates and some microlocal analysis tools such as microlocaldefect measures. We include an appendix on the latter topic here to make thearticle self contained and supplement details to proofs of some of the theoremswhich can be found in the lecture notes of Burq and G\'erard (2001). Once weestablish essential observability properties for the approximate solutions, itis not difficult to prove that the solution of the original problem alsopossesses a similar feature via a delicate passage to limit. In the last partof the paper, we establish various decay rate estimates for different growthconditions on the nonlinear dissipative effect. We in particular generalize theknown results on the subject to a considerably larger class of dissipativeeffects.

Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation