Abstract
A diagonal system of three first-order partial differential equations in two independent variables is considered.The equations entering into the diagonal system are independent from each other, therefore, the compatibility condition of the system does not arise.We consider the asymptotic behavior of solutions at an infinitely distant point, with respect to some parameter.The main place in the system is occupied by a nonlinear first-order partial differential equation, the remaining equations are adjoining equations, the solutions of which contain the initial value of one independent variable as a parameter.The attached equations are chosen appropriately, and the solution to the system is already studied, which already has an internal connection.The adjoint equations are linear first-order partial differential equations.Using the fact that the zero solutions of the characteristic equations are asymptotically stable on Lyapunov, the conditions when the set of three differential equations, considered as a diagonal system of partial differential equations of the first order, has a solution with certain initial values and is an infinitesimal function in the vicinity of an infinitely remote point are described .Methods of the theory of functions and differential inequalities in the theory of first-order differential equations are used.