In this work, we consider a Cauchy problem for the generalized Schrö dinger equation which has important applications in quantum kinetic theory, water wave problems and ferromagnetism. Due to its multidimensionality, it is important from the point of view of modern physics theories such as quantum field theory and string theory. We prove the uniqueness of the solution of the problem in an unbounded domain by using semigeodesic coordinates. The main tool is a pointwise Carleman estimate. To the authors' best knowledge, this is the first study which deals with the solvability of this problem.

In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering, chemotaxis, and immunology.