Appl Math Optim 51:61–105 (2005)
2004 Springer Science+Business Media, Inc.
Convexity and Weighted Integral Inequalities for Energy
Decay Rates of Nonlinear Dissipative Hyperbolic Systems
L.M.A.M., CNRS-UMR 7122, Universit´e de Metz,
Ile du Saulcy, 57045 Metz Cedex 01, France
Communicated by I. Lasiecka
Abstract. This work is concerned with the stabilization of hyperbolic systems by
a nonlinear feedback which can be localized on a part of the boundary or locally
distributed. We show that general weighted integral inequalities together with con-
vexity arguments allow us to produce a general semi-explicit formula which leads to
decay rates of the energy in terms of the behavior of the nonlinear feedback close to
the origin. This formula allows us to unify for instance the cases where the feedback
has a polynomial growth at the origin, with the cases where it goes exponentially fast
to zero at the origin. We also give three other signiﬁcant examples of nonpolynomial
growth at the origin. Our work completes the work of  and improves the results
of  and  (see also  and ). We also prove the optimality of our results
for the one-dimensional wave equation with nonlinear boundary dissipation. The
key property for obtaining our general energy decay formula is the understanding
between convexity properties of an explicit function connected to the feedback and
the dissipation of energy.
Key Words. Nonlinear dissipation, Boundary damping, Locally distributed feed-
back, Hyperbolic equations, Optimality.
AMS Classiﬁcation. 34G10, 35B35, 35B37, 35L90, 93D15, 93D20.
In this paper we are interested in energy decay formulas for hyperbolic systems subjected
to nonlinear dissipation with arbitrary growth conditions at the origin. Before stating our