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On the determination of nonlinear terms appearing in semilinear hyperbolic equations
Author(s) -
Kian Yavar
Publication year - 2021
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12440
Subject(s) - dimension (graph theory) , nonlinear system , riemannian manifold , manifold (fluid mechanics) , mathematics , boundary (topology) , hyperbolic partial differential equation , term (time) , mathematical analysis , inverse , class (philosophy) , boundary value problem , pure mathematics , partial differential equation , physics , geometry , mechanical engineering , quantum mechanics , artificial intelligence , computer science , engineering
Abstract We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary ( M , g ) of dimension n = 2 , 3 . We prove results of unique recovery of the nonlinear term F ( t , x , u ) , appearing in the equation∂ t 2 u − Δ g u + F ( t , x , u ) = 0 on ( 0 , T ) × M with T > 0 , from partial knowledge of the solutions u on the lateral boundary ( 0 , T ) × ∂ M . We obtain, what seems to be, the first result of determination of the expression F ( t , x , u ) on the boundary x ∈ ∂ M for such a general class of nonlinear terms. With additional assumptions on the manifold and some extended measurements at t = 0 and t = T , we prove also the recovery of F inside the manifold x ∈ M .