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. know result about Polynomial stability.
Jaime E. Muñoz Rivera
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
A sufficient and a necessary condition to polynomial decay was given by J. Pruss 2006
Other importante result due to Pruss 2006 was given by J. 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
Our main result is the following necessary condition given by a result due to
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.
Aplications given by a result due to
The Infenitesimal generator of the semigroup is given by a result due to
We denote the associated semigroup as
Previous given by a result due toresult
That system was studied by Chen and Triggiani. They given by a result due to 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.
and Differentiable when
Our stability result to damped wave equation is
to Bresse system