Optimality on Polynomial decay of semigroups in

Elasticity

By

Jaime E. Muñoz Rivera

LNCCIM-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

Our purporse is to show that the sufficient condition of Liu and Rao is also a necessary condition

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.

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 LondrinaParaná – Brasil e-mail: lucifatori@uel.br

Our interest is to prove that the sufficient condition of Liu and Rao is also a necessary condition.

This result will be important to show when a rate of decay is optimal.

Our main result is the following necessary condition

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:

Jaime E M Rivera

Faltó resumen y conclusiones.