**Mourad Choulli**^{[a]}

*Various stability estimates for
the problem of determining an initial
heat distribution from a single measurement
*

**Pages:** 279-307

**Received:** 22 December 2015

**Accepted in revised form:** 24 January 2017

**Mathematics Subject Classification (2010):** 35R30.

**Keywords:** Heat equation, fractional heat equation, initial heat distribution, Müntz's theorem, point measurement, boundary measurement,
stability estimates of Hölder and logarithmic type.

**Author address:**

[a]: Université de Lorraine, Institut Élie Cartan de Lorraine,UMR CNRS 7502, Boulevard des Aiguillettes, BP 70239
54506 Vandoeuvre les Nancy cedex - Ile du Saulcy
57045 Metz cedex 01, France

**Abstract:**
We consider the problem of determining the initial heat
distribution in the heat equation from a point measurement. We show
that this inverse problem is naturally related to the one of recovering
the coefficients of Dirichlet series from its sum. Taking the advantage
of existing literature on Dirichlet series, in connection with Müntz's
theorem, we establish various stability estimates of Holder and logarithmic type. These stability estimates are then used to derive the
corresponding ones for the original inverse problem, mainly in the case
of one space dimension.
In higher space dimensions, we are interested to an internal or a boundary measurement. This issue is closely related to the problem of observability arising in Control Theory. We complete and improve the
existing results.

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