Abstract
This work is concerned with the study of the inverse problem of determining two coefficients in a hyperbolic-parabolic system using the following observation data: an interior measurement of only one component and data of two components at a fixed time over the whole spatial domain. A Lipschitz stability result is proved using Carleman estimates.
A Appendix
Proof of Lemma 2.
Let us start with the estimation of
Then, we obtain
After an integration by parts, we get
Moreover, for all
Using the formula
Substituting this estimate into (A.2), we have
On the other hand, for
Integrating this inequality for t between
Finally, we have
Now, we want to estimate
Thus we have
which gives
Integrating between η and
Finally, thanks to (A.4) and (A.3), we obtain
That is
Next, we will estimate
Lemma 1.
Let
We have
Then
An integration by parts leads to
On the other hand, we have
Thus
Applying Lemma 1 to the last inequality, we obtain
For s large enough, the last term of the right hand side is absorbed by the last term of the left hand side, so we have
This leads, for s large, to the following inequality:
According to the assumption of Lemma 2 we obtain
In the same way as for
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