---

Daniel Toundykov (University of Nebraska)

Title: Energy dissipation in dynamical systems governed by nonlinear PDEs

Abstract: Controllability, uniform stability, as well as properties of global attractors of infinite-dimensional dissipative dynamical systems can often be established by studying appropriate classes of observability estimates. Of a particular interest is the fact that the stability of a linear problem may imply the corresponding observability inequality for the analogous non-linear system. This relation will be demonstrated on the example of a semilinear wave equation with locally distributed boundary damping. In the linear case the energy of the system decays exponentially with time, whereas in the nonlinear setting the dissipation rate is described by the solution to a nonlinear ODE which can be directly constructed from the feedback map. This approach essentially reduces the study of a nonlinear PDE to a linear PDE and a nonlinear ODE. Moreover, this method permits to analyze saturated or superlinear (at infinity) dissipative feedbacks, and exhibits the relation between the rate of the energy decay and the regularity of solutions. The framework can also be extended to any dissipative dynamical system which obeys suitable energy inequalities.