Appl Math Optim (2014) 69:83–122
Uniﬁed Approach to Stabilization of Waves on Compact
Surfaces by Simultaneous Interior and Boundary
Feedbacks of Unrestricted Growth
Marcelo M. Cavalcanti ·
Valéria N. Domingos Cavalcanti ·
Ryuichi Fukuoka · Daniel Toundykov
Published online: 12 September 2013
© Springer Science+Business Media New York 2013
Abstract Let M ⊂ R
be an oriented compact surface on which we consider the
) = 0inM×]0,∞[,
u+ u+ b(x)g(u
) = 0on∂M×]0,+∞[.
If M along with the localizers a, b and the nonlinear feedbacks g, g
conditions then uniform (but not necessarily exponential) decay rates of the ﬁnite
energy of solutions can be established.
We present a uniﬁed approach that bridges and extends a number of earlier results
on stabilization of 2nd-order hyperbolic equations on manifolds. The methodology
captures geometric requirements for damping acting simultaneously on subsets of
the interior and of the boundary, and shows how placements of these feedbacks can
complement each other depending on the underlying surface. In addition, the results
Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0.
Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant
Research of Ryuichi Fukuoka is partially supported by the CNPq Grant 305557/2009-2.
Research of Daniel Toundykov is partially supported by the NSF Grant DMS-0908270.
M.M. Cavalcanti · V.N.D. Cavalcanti · R. Fukuoka
Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil
D. Toundykov (
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, USA