Stability of the determination of a time-dependent coefficient in parabolic equations

Mourad Choulli; Yavar Kian

doi:10.3934/mcrf.2013.3.143

Article Contents

2013, Volume 3, Issue 2: 143-160. Doi: 10.3934/mcrf.2013.3.143

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Stability of the determination of a time-dependent coefficient in parabolic equations

Mourad Choulli^1, and
Yavar Kian^2,

1.
LMAM, UMR 7122, Université de Lorraine, Ile du Saulcy, 57045 Metz, cedex 1
2.
UMR-7332, Aix Marseille Université, Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille, cedex 9

Received: January 31, 2012

Revised: April 30, 2012

Published: May 2013

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Abstract

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.

Keywords:

Parabolic equation,
semi-linear parabolic equation,
inverse problem,
determination of time-depend coefficient,
stability estimate.

Mathematics Subject Classification: 35R30.

Citation:

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References

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