TY - JOUR
T1 - On determining and breaking the gauge class in inverse problems for reaction-diffusion equations
AU - Kian, Yavar
AU - Liimatainen, Tony
AU - Lin, Yi Hsuan
N1 - Publisher Copyright:
© The Author(s), 2024. Published by Cambridge University Press.
PY - 2024/2/26
Y1 - 2024/2/26
N2 - We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on Rn. Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.
AB - We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on Rn. Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.
UR - http://www.scopus.com/inward/record.url?scp=85186869077&partnerID=8YFLogxK
U2 - 10.1017/fms.2024.18
DO - 10.1017/fms.2024.18
M3 - Article
AN - SCOPUS:85186869077
SN - 2050-5094
VL - 12
JO - Forum of Mathematics, Sigma
JF - Forum of Mathematics, Sigma
M1 - e25
ER -