Second order systems on Hilbert spaces with nonlinear damping

Abstract

We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems out of thin air introduced by M. Tucsnak and G. Weiss in 2003. The modified differential equation contains a new nonlinear damping term, that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.