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Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJPowerPoint Presentation

Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ

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Presentation Transcript

Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that

knowresultabout

Polynomialstability

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

This is a join work with Luci Fatori: given by a result due to

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 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.

Z. Liu and K. Liu, proved that the semigroup is analytic when

and Differentiable when

Our when

Contribution

Our contribution is about polynomial stability for when

Our stability result to damped wave equation is

Idea whenoftheproofoftheoptimality:

Thanks when

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