Exact solution to one functional differential equation of parabolic type using semigroup theory and some of its applications

In this paper, the Cₒ-semigroup operator is constructed, the infinitesimal generator of which is a linear combination of the diffusion-thermal conductivity operator and the spatial inversion operator on a straight line. The general properties of the evolution of functions defining initial conditions under the action of such an operator are discussed. It is shown that these properties differ sharply from the properties of solutions to a one-dimensional parabolic equation on a straight line due to the presence of an essentially discrete element in the generator of the semigroup under consideration.