Inverse Problems for Time-Dependent Singular Heat Conductivities---One-Dimensional Case

International audienceWe 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) = k 2 (0 < x < s(t)), γ(t, x) = 1 (s(t) < x < 1). Firstly 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). Secondly 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)