S. Li
Estimation of coefficients in a hyperbolic equation with impulsive inputs
For the solution to
u(x, t) – Δu(x, t) + q(x)u(x, t) = δ(x
1)δ′´(t) and u|
t<0 = 0, we consider an inverse problem of determining q(x), x ∈ Ω from data ƒ = u|ST
and g = (∂u/∂v)|ST
. Here Ω ⊂ {(x
1, . . . , x
n
) ∈ |x
1 > 0}, n ≥ 2, is a bounded domain, S
T
= {(x, t) | x ∈ ∂Ω, x
1 < t < T + x
1} and T > 0. For suitable T > 0, we prove an L
2 (Ω)-size estimation of q:
||q||L
2(Ω) ≤ C{||ƒ||H
1(S
T
) + ||g||L
2(ST
)},
provided that q satisfies a priori uniform boundedness conditions.
We use an inequality of Carleman type in our proof.
Journal of Inverse and Ill-posed Problems, Walter de Gruyter
Print ISSN: 0928-0219
Volume: 14, 12/2006
Pages: 891 - 904
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