TY - JOUR
T1 - Inverse problems for first-order hyperbolic equations with time-dependent coefficients
AU - Floridia, Giuseppe
AU - Takase, Hiroshi
N1 - Funding Information:
This work was supported in part by Grant-in-Aid for JSPS Fellows Grant Number JP20J11497 and Istituto Nazionale di Alta Matematica (INδAM), through the GNAMPA Research Project 2020, titled “Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni”, coordinated by the first author. Moreover, this research was performed in the framework of the French-German-Italian Laboratoire International Associé (LIA), named COPDESC, on Applied Analysis, issued by CNRS, MPI, and INδAM, during the INδAM Intensive Period-2019, “Shape optimization, control and inverse problems for PDEs”, held in Napoli in May-June-July 2019.
Funding Information:
This work was supported in part by Grant-in-Aid for JSPS Fellows Grant Number JP20J11497 and Istituto Nazionale di Alta Matematica (IN?AM), through the GNAMPA Research Project 2020, titled ?Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni?, coordinated by the first author. Moreover, this research was performed in the framework of the French-German-Italian Laboratoire International Associ? (LIA), named COPDESC, on Applied Analysis, issued by CNRS, MPI, and IN?AM, during the IN?AM Intensive Period-2019, ?Shape optimization, control and inverse problems for PDEs?, held in Napoli in May-June-July 2019. This paper owes much to the thoughtful and helpful comments of Professor Piermarco Cannarsa (University of Rome ?Tor Vergata?) and Professor Masahiro Yamamoto (The University of Tokyo). In particular, we thank Prof. Cannarsa to have suggested to us that the inequality (2.5) implies a precise exponential structure for the coefficient A(x,t) (that we showed in Proposition 2.10). Moreover, we are extremely grateful to Prof. Yamamoto to have fully read a draft of this paper and to have given us a lot of pieces of advice to make it more clear and readable.
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/12/25
Y1 - 2021/12/25
N2 - We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.
AB - We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.
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U2 - 10.1016/j.jde.2021.10.007
DO - 10.1016/j.jde.2021.10.007
M3 - Article
AN - SCOPUS:85117182033
SN - 0022-0396
VL - 305
SP - 45
EP - 71
JO - Journal of Differential Equations
JF - Journal of Differential Equations
ER -