Graduate School of Mathematical Sciences Komaba, Tokyo, Japan Inverse Hyperbolic Problem by a Finite Time of Observations with Arbitrary Initial Values (original) (raw)

We consider a solution u(p, g, a, b) to an initial value-boundary value problem for a hyperbolic equation: ∂ t u(x, t) = ∆u(x, t) + p(x)u(x, t), x ∈ Ω, 0 < t < T u(x, 0) = a(x), ∂tu(x, 0) = b(x), x ∈ Ω, u(x, t) = g(x, t), x ∈ ∂Ω, 0 < t < T. and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T ) and u(p, g, a, b)(x, T0), ∂tu(p, g, a, b)(x, T0), x ∈ Ω with T0 < T . We prove that if T −T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, ..., gN by the Hilbert Uniqueness Method, so that the mapping {u(p, gj , aj , bj)|ω×(0,T ), u(p, gj , aj , bj)(·, T0), ∂tu(p, gj , aj , bj)(·, T0)}1≤j≤N −→ {p, aj , bj}1≤j≤N is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj , bj}1≤j≤N remains some bounded set in a suitable Sobolev space. In our inverse problem, initial values are a...