Article

# Inverse Hyperbolic Problems with Time-Dependent Coefficients

Communications in Partial Differential Equations (impact factor: 0.89). 11/2007; 32:1737-1758. DOI:10.1080/03605300701382340 pp.1737-1758
Source: arXiv

ABSTRACT We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.

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##### Article:Stability of the determination of a time-dependent coefficient in parabolic equations
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ABSTRACT: We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation in changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(t,x,\sigma(t),u(x,t))$.
02/2012;

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### Keywords

BLR condition

by-product

hyperbolic equation

inverse problem

lower order terms

similar result

time-dependent Dirichlet-to-Neumann operator

time-independent coefficients