Special Structure of the Solution to the Cauchy Problem for a Parabolic Equation and Inverse Problems

For an equation of parabolic type whose principal part is the heat operator, we study the Cauchy problem with a point source. A special structure of the solution to this problem is written out. It is based on a representation of the solution as the product of the fundamental solution to the heat equation and a polynomial in powers of $ t $ with coefficients depending on the spatial variables. Formulas for computing these coefficients are derived, and an estimate of the remainder term is given. Next, two inverse problems for the original equation are posed. They are then studied on the basis of the obtained structure of the solution to the Cauchy problem. A uniqueness theorem is formulated for the considered inverse problems.