Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide

ABSTRACT

We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.

Keywords:

• Inverse problem
• heat equation
• time-dependent source term
• stability inequality
• Carleman estimate
• partial boundary data

2010 Mathematics subject classifications:

• 35R30
• 35K05

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This is where the assumption that $\mathrm{\partial \Omega }$ be ${\mathcal{C}}^4$ is needed.

Additional information

Funding

D. Sambou is supported by the Chilean Fondecyt [grant number 3170411]. YK and ÉS are partially supported by the Angence Nationale de la Recherche (ANR) [grant number ANR-17-CE40-0029].