INVERSE PARABOLIC PROBLEMS BY CARLEMAN ESTIMATES WITH DATA TAKEN AT INITIAL OR FINAL TIME MOMENT OF OBSERVATION

. We consider a parabolic equation in a bounded domain & omega; over a time interval (0, T) with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary & UGamma; & SUB; partial differential & omega;. Then, we consider an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution u: u|(0,T )x & UGamma; and u(t0, & BULL;) in & omega; with t0 = 0 or t0 = T. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case t0 = T. Second, under additional condition for & UGamma;, we prove the uniqueness for the case t0 = 0. The second result adjusts the uniqueness by M.V. Klibanov (Inverse Problems 8 (1992) 575-596) to the inverse problem in a bounded domain & omega;. 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 t0 = 0 without boundary condition on the whole partial differential & omega;.

Automated summary

This paper investigates a parabolic equation in a bounded domain & omega; over a time interval (0, T) with homogeneous Neumann boundary condition. It focuses on an inverse problem of determining a zeroth-order spatially varying coefficient through additional data of the solution u: u|(0,T )x & UGamma; and u(t0, & BULL;) in & omega; with t0 = 0 or t0 = T. The paper establishes a conditional Lipschitz stability estimate and uniqueness for the case t0 = T, and proves the uniqueness for the case t0 = 0 under additional conditions for & UGamma;. The uniqueness result adapts the method of M.V. Klibanov (Inverse Problems 8 (1992) 575-596) to the inverse problem in a bounded domain & omega;, by modifying the inverse parabolic problem to an inverse hyperbolic problem.