Special examples of superstable semigroups and their application in the inverse problems theory

Special examples of superstable (quasinilpotent) semigroups and their application in the theory of linear inverse problems for evolutionary equations are studied. The term “semigroup” means here the semigroup of bounded linear operators of class $C_0$. The standard research scheme is used. The linear inverse problem with the final overdetermination in a Banach space for the evolution equation is considered. A special assumption is introduced, related to the superstability of the main evolutionary semigroup. For the inverse problem we establish the existence and uniqueness theorem of the solution. It is noted that the solution of the problem can be represented by the convergent Neumann series. To illustrate the general theory, we consider special examples of superstable semigroups that are generated by a one-dimensional streaming operator with absorption in the weighted Banach space of functions on the ray. It is shown that there are many possibilities for choosing the absorption coefficient and the weight function, under which the superstability of the corresponding semigroup is guaranteed. The established results allow applying to a particular inverse problem for the transport equation with absorption on the ray. The applied approach can be extended to the multidimensional transport equation in an unbounded domain without the collision integral.