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On Reconstruction of the Diffusion and of the Principal Coefficient of a Hyperbolic Equation

  • Victor Isakov
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 90)

Abstract

Consider the second order hyperbolic equation
$${u_{{tt}}} - div\left( {a\nabla u} \right) + b{u_{t}} = 0{\text{ }}in{\text{ }}Q = \times \left( {0,T} \right)$$
(1.1)
with zero initial conditions
$$u = {u_{t}} = 0{\text{ }}on{\text{ }}\Omega \left\{ 0 \right\}$$
(1.2)
and the lateral Neumann boundary condition
$$a{\partial _{v}}u = h{\text{ }}on{\text{ }}\partial \Omega \times \left( {0,T} \right)$$
(1.3)
Under rather general assumptions: a, 6 do not depend on x, are in L∞(Ω), a is C 1 near ∂Ω, and > ∈ > 0 this initial boundary value problem has a unique solution u, u t ∈ H 1 2 (Q) provided h, h t ∈ L2(öΩ × (0, T)), h = 0 on öΩ × {0}, and ∞ is a bounded domain in ℝ n with the boundary öΩ ∈ C 1 or fi is a half-space in ℝ n (and then h is assumed to be compactly supported). A solution u is understood in the generalized sense as a function satisfying the following integral equality \(\int\limits_{Q} {\left( {{u_{t}}{v_{t}} - a\nabla u \cdot \nabla v - b{u_{t}}v} \right)dx{\text{ }}dt = {\text{ }}\int\limits_{{\partial \Omega \times \left( {0,T} \right)}} {h{\text{ }}v{\text{ }}dS\left( {x,t} \right)} }\) for any function v from the same class as u but satisfying the zero Cauchy data on ∞ × {T}. In geophysics of fundamental importance is the following inverse problem: find a,b given either for one h (single boundary measurement) or for all (regular) h supported in F × (0,T). Here Γ is a part of öΩ which can coincide with öΩ, but in most of geophysical applications it is relatively small part of it. In fact, F is the observation area.

Keywords

Inverse Problem Wave Equation Cauchy Problem Hyperbolic Equation Plane Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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