Various stability estimates for the problem of determining an initial heat distribution from a single measurement. (English) Zbl 1379.35349

Choulli, Mourad

Riv. Mat. Univ. Parma (N.S.) 7, No. 2, 279-307 (2016).

Summary: 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 Hölder 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 dimention.
In higher space dimesions, we are interested to a 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.

MSC:

35R30

Inverse problems for PDEs

Keywords:

heat equation; fractional heat equation; initial heat distribution; Müntz’s theorem; point measurement; boundary measurement; stability estimates of Hölder and logarithmic type