On the solvability of the identification problem for a source function in a quasilinear parabolic system of equations in bounded and unbounded domains

The paper considers the problem of identification for a source function in one of two equations of parabolic quasilinear system. The case of Cauchy data in an unbounded domain and the case of boundary conditions of the first kind in a rectangular domain are considered. The question of the existence and uniqueness of the solution is studied. The proof uses a differential level splitting method known as the weak approximation method. The solution is obtained on a small time interval in the class of sufficiently smooth bounded functions.