Inverse problems for time-dependent singular heat conductivities?one- dimensional case

Abstract

We consider an inverse boundary value problem for the heat equation on the interval (0, 1), where the heat conductivity γ(t, x) is piecewise constant and the point of discontinuity depends on time: γ(t, x) = k2 (0 < x < s(t)), γ(t, x) = 1 (s(t) < x < 1). First, we show that k and s(t) on the time interval [0, T] are determined from a partial Dirichlet-to-Neumann map: u(t, 1) → ∂xu(t, 1), 0 < t < T, u(t, x) being the solution to the heat equation such that u(t, 0) = 0, independently of the initial data u(0, x). Second, we show that another partial Dirichlet-to-Neumann map: u(t, 0) → ∂xu(t, 1), 0 < t < T, u(t, x) being the solution to the heat equation such that u(t, 1) = 0, restricts the pair (k, s(t)) to, at most, two cases on the time interval [0, T], independently of the initial data u(0, x). © 2013 Society for Industrial and Applied Mathematics.