Optimality on Polynomial
decay of semigroups in
Elasticity
By
Jaime E. Muñoz Rivera
LNCC
IM-UFRJ
Our problem:
Given a semigroup of contractions T(t) defined
over Hilbert space H, find the best number p such
that
Liu and Rao proved a sufficient
condition to get the polynomial
decay of a semigroup
Removing logarits, Liu and Rao’s Theorem can be
written as:
A sufficient and a necessary
condition to polynomial decay was
given by J. Pruss 2006
Other importante result due to Pruss 2006
Pruss method give a necessary and a sufficient
condition to prove polynomial estability
The problem is that it is not a simple task to estimate
fractional powers of the operator of I.G.S.
It is more easy to deal with the sufficient condition
given by Liu and Rao.
Our purporse is to show that the sufficient condition
of Liu and Rao is also a necessary condition
The nexus between Liu-Rao and Pruss Characterization
is given by a result due to
This is a join work with Luci Fatori:
Estadual University of Londrina
Paraná – Brasil
e-mail: [email protected]
Our interest is to prove that the sufficient condition of Liu
and Rao is also a necessary condition.
Our main result is the following necessary condition
This result will be important to show when a rate of decay
is optimal.
The proof is based on Pruss necessary condition and
Latushkin –Shvidkoy result.
The Infenitesimal generator of the semigroup is
We denote the associated semigroup as
That system was studied by Chen and
Triggiani. They proved that the
semigroup is analytic if
The authors solved the conjetures of G.
Chen and D. L. Russel on structural
damping for elastic systems.
Z. Liu and K. Liu, proved that the semigroup
is analytic when
and Differentiable when
Our contribution is about polynomial stability for
Our stability result to damped wave equation is
Idea of the proof of the optimality: