A new approach of stabilization of nondissipative distributed systems

In this paper we propose a new approach to prove the nonlinear (internal or boundary) stabilization of certain nondissipative distributed systems (the usual energy is not decreasing). This approach leads to decay estimates (known in the dissipative case) when the integral inequalities method due to Komornik [Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, John Wiley, Chichester, UK, 1994] cannot be applied due to the lack of dissipativity. First we study the stability of a semilinear wave equation with a nonlinear damping based on the equation u-Deltau+h(delu)+f(u)+g(u')=0. We consider the general case with a function h satisfying a smallness condition, and we obtain uniform decay of strong and weak solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related with the above equation. In the second part we consider the case h(delu)= -delphi.delu with phi is an element of W-1,W-infinity(Omega). We prove some precise decay estimates (exponential or polynomial) of equivalent energy without any restriction on phi. The same results will be proved in the case of boundary feedback. Finally, we comment on some applications of our approach to certain nondissipative distributed systems. Some results of this paper were announced without proof in [A. Guesmia, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 633-636].