Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

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

Pages: 279-307
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.
[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 Holder 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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