Uniqueness of the Solution of Inverse Initial Value Problems for the Burgers Equation with an Unknown Source

We consider inverse problems of determining the initial conditions and a time-invariant inhomogeneity in boundary value problems for the Burgers equation. A transformation is used permitting one to reduce the Burgers equation to an equation with a variable coefficient for a function available for measurement in tomographic observations. We prove theorems on the unique reconstruction of the initial conditions based on the mean values of the solution with respect to time or space variables. Inverse problems of the simultaneous determination of the initial data and the source on an interval and a half-line are stated. The uniqueness of their solutions is proved on the basis of spectral representations.