An Inverse Problem for a Nonlinear Transport Equation with Final Overdetermination

In this paper, we are interested in the nonlinear transport equation in the class of essentially bounded functions. More precisely, we prove the existence and uniqueness of the inverse problem solution. Moreover, we give the size of the neighborhood from which we can bring the function of the overdetermination condition so that the solution of the inverse problem is unique. The method is based on known properties of solutions of the (direct and inverse) linear problems and uses twice the inverse function theorem in the corresponding function spaces. Then we use the refined inverse function theorem to obtain the sufficient conditions for the unique solvability of the initial nonlinear inverse problem in terms of the size of the neighborhood from which we can bring the function of the overdetermination condition.