2022/11/22 by O. Imanuvilov, M. Yamamoto, Imanuvilov, O. +1
Computer Science · Mathematics · #35R30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2211.11930
openalex publication_date 2022/11/22 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
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 Γ⊂ \ppp\OOO. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution u: u\vertΓ× (0,T) and u(⋅,t0) in \OOO with t0=0 or t=T. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case t0=T. Second, under additional condition for Γ, we prove the uniqueness for the case t0=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 t0=0 without boundary condition on the whole \ppp\OOO.