Structure of solutions for model inverse problem for the heat equation in classes of exponential growth functions

Model inverse problem of finding the inhomogeneous term of the one-dimensional heat equation is considered. The research is performed in classes of exponential growth functions. The condition of uniqueness of solution is established. It is shown that in case of breakdown of this condition any non-trivial solution of exponential growth can be represented as a linear combination of elementary solutions. Examples of special solutions with superexponential growth are given.