Abstract :
The problem of obtaining uniform decay rates for linear and nonlinear boundary value problems with boundary dissipation has been largely studied in the last years in the context of boundary control theory where the existence of stabilizing boundary feedback is important. However, the problems were studied on a case by case basis, only for some particular types of boundary feedbacks and moreover, it was not clear to which extent nonlinear conservative terms and nonlinear boundary feed-backs can be allowed. Our goal here is to give a general approach for such problems, which is valid for all evolution P.D.E of hyperbolic or ultra-hyperbolic type. In addition, under stronger assumptions we consider the problem of existence of compact attractors. This work is essentially based on some a-priori estimates of Carleman′s type obtained by the author in a previous paper [27].
