Determination of the Memory from Boundary Measurements on a Finite Time Interval

We study the problem of finding the memory term of a hyperbolic equation from the Dirichlet-to-Neumann map given on a finite time interval. We prove that this map determines uniquely some characteristics of the memory function and thereby memory functions of a special form. 1. STATEMENT OF THE PROBLEM AND THE MAIN RESULT Consider the equation utt −∆u + ∫ t 0 k(x, t− τ)u(x, τ) dτ = 0 (1.1) in Ω× [0,∞), where k(x, t) ∈ C∞(Ω× [0,∞)) and Ω is a bounded domain in R. We pose the initial-boundary value problem, by supplementing (1.1) with the conditions u(x, 0) = ut(x, 0) = 0, (1.2) u|∂Ω×[0,∞) = g(x, t), (1.3) where g ∈ C∞ 0 (∂Ω × [0,∞)), the subscript 0 means that g vanishes at t = 0 and for t > T together with its derivatives, with T some positive number, which guarantees validity of the agreement conditions. Existence and uniqueness of a solution to the problem ensues from Theorem 1.2 below. Assume that we can choose various functions g ∈ C∞ 0 (∂Ω × [0,∞)) and measure the normal derivative ∂u/∂ν on the set Γ = ∂Ω×[0, T ], T > 0. Observe that ∂u/∂ν on Γ depends only on the values of g on Γ; so we can assume that ∗Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Acad. Koptyug prosp. , 4, 630090, Novosibirsk, Russia. E-mail: dyatlov@math.nsc. ru The work was supported by RFBR grant No. 99–01–00540. Determination of the Memory 9 g = 0 for t > T ; i.e., g ∈ C∞ 0 (Γ). The inverse problem consists in finding the memory k(x, t) from the addition information {(g, ∂u/∂ν|Γ) | g ∈ C∞ 0 (Γ)}. The main result of the article is the following theorem: Theorem 1.1. Let k1 and k2 be two functions satisfying the above conditions. Suppose that T > diam Ω. If ∂u1/∂ν = ∂u2/∂ν on Γ for all g ∈ C∞ 0 (Γ), where uj are solutions to (1.1)–(1.3) with k = kj , then ∂ t k1(x, 0) = ∂ m t k2(x, 0), m = 0, 1, 2. In the case T = ∞ we problem under consideration was studied in [1], wherein a conditional stability estimate of the logarithmic type was proven. The proof was based on the reduction of the original problem to a family of stationary problems with a parameter by means of the Fourier transform in time. In the case of a finite time interval this method is obviously inapplicable. A similar problem for the equation utt −∆u + q(x)u = 0 was considered by Rakesh and Symes [2] who proved a uniqueness theorem. Stability was proven by Sun [5]. The case of time-dependent q was studied in [3, 4]. For proving Theorem 1.1 we actually use the method of beam solutions proposed in [2]. The difference is that the problem in [2] reduces to the ray transform, while here we reduce the problem to the Fourier transform. Below we need a solvability result for the direct problem. Theorem 1.2. The problem utt −∆u + ∫ t 0 k(x, t− τ)u(x, τ) dτ = f(x, t), (x, t) ∈ Ω× [0,∞), u(x, 0) = ut(x, 0) = 0, x ∈ Ω, u|∂Ω×[0,∞) = g(x, t), x ∈ ∂Ω, 0 ≤ t < ∞, where f ∈ C∞(Ω × [0,∞)) is bounded and g ∈ C∞ 0 (Γ), has a unique solution u ∈ C∞(Ω× [0,∞)). Moreover, the estimate ‖u‖H1(Ω×[0,T ]) ≤ C ( ‖f‖L2(Ω×[0,T ]) + ‖g‖H1(Γ) ) , holds with some positive constant C depending only on Ω and k. The assertion can be derived from the general theory of the initial-boundary value problems for hyperbolic equations as it was done, for example, in [1]. 2. AUXILIARY ASSERTIONS Henceforth we denote the convolution ∫ t 0 k(x, t− τ)u(x, τ) dτ by k ∗ u.