An Inverse Problem for the Heat Equation

Let ut = uxx − q x u 0 ≤ x ≤ 1 t > 0 u 0 t = 0 u 1 t = a t u x 0 = 0, where a t is a given function vanishing for t > T a t ≡ 0 T 0 a t dt < ∞. Suppose one measures the flux ux 0 t = b0 t for all t > 0. Does this information determine q x uniquely? Do the measurements of the flux ux 1 t = b t give more information about q x than b0 t does? These questions are answered in this note.