Appl Math Optim 48:211–228 (2003)
2003 Springer-Verlag New York Inc.
Global Uniqueness and Stability for a Class of Multidimensional
Inverse Hyperbolic Problems with Two Unknowns
and Xu Zhang
Department of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba Meguro, Tokyo 153, Japan
School of Mathematics, Sichuan University,
Chengdu 610064, People’s Republic of China
Departamento de Matem´aticas, Facultad de Ciencias,
Universidad Aut´onoma de Madrid,
28049 Madrid, Spain
Communicated by I. Lasiecka
Abstract. In this paper we obtain the global uniqueness and stability estimate for a
class of multidimensional inverse hyperbolic problems of determining a source term
and an initial value from a single measurement of boundary values or interior values.
By means of a suitable transformation, we reduce the problem to the observability
inequalities for nonconservative hyperbolic equations with memory. Then, using a
compactness/uniqueness argument, we can prove the uniqueness and the stability
by a new kind of unique continuation property of a nonlocal hyperbolic equation.
Key Words. Global uniqueness, Stability estimate, Observability, Inverse hyper-
bolic problem, Two unknowns.
AMS Classiﬁcation. Primary 35R30, Secondary 74J25, 93B07.
The second author was partially supported by the Foundation for the Author of National Excellent
Doctoral Dissertation of China (Project No. 200119), Grant BFM2002-03345 of the Spanish MCYT, and the
National Natural Science Foundation of China.