Inverse parabolic problems by Carleman estimates with data taken initial or final time moment of observation

TL;DR

This paper establishes uniqueness and stability results for an inverse parabolic problem, determining a spatial coefficient from partial boundary and interior data, by adapting Carleman estimates and reducing to hyperbolic problems.

Contribution

It extends Klibanov's method to bounded domains, providing new uniqueness results for inverse parabolic problems with partial data and no boundary conditions.

Findings

Conditional Lipschitz stability for $t_0=T$ case.

Uniqueness under additional conditions for $t_0=0$ case.

Proved uniqueness for inverse source problem without boundary conditions.

Abstract

We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $\Gamma \subset \ppp\OOO$. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution $u$: $u\vert_{\Gamma \times (0,T)}$ and $u(\cdot,t_0)$ in $\OOO$ with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case $t_0=T.$ Second, under additional condition for $\Gamma$, we prove the uniqueness for the case $t_0=0$. The second result adjusts the uniqueness by M.V. Klibanov (Inverse Problems {\bf 8} (1992) 575-596) to the inverse problem in a bounded domain $\OOO$. We modify his method which reduces the inverse parabolic problem to an inverse hyperbolic problem, and so even for the inverse parabolic problem, we have to assume conditions for the uniqueness for the corresponding inverse hyperbolic problem. Moreover we prove the uniqueness for some inverse source problem for a parabolic equation for $t_0=0$ without boundary condition on the whole $\ppp\OOO$.