On a diagonal system of the first-order partial differential equations from two independent variables

Abstract

A diagonal system of three first-order partial differential equations in two indep endent variables
is cons idered. The equations entering into the diagonal system are indep endent from each other,
therefore, the compatibility condition of the system do es not arise. We consider the asymptotic b ehavior of solutions at an infinitely distant p oint, with resp ect to some parameter. The main place
in the system is o ccup ied by a nonlinear first-order partial differential equation, the remaining
equations are adjoining equations, the solutions of which contain the initial value of one indep endent variable as a parameter. The attached equations are chosen appropriately, and the solution
to the system is already studied, w hich already has an internal connection. The adjoint equations
are linear first-orde r partial differential equations. Using the fact that the zero solutions of the
characteristic equations are asymptotically stable on Lyapunov, the condi tion s when the set of
three differential equations, c on sidered as a diagonal system of partial differential equations of the
first order, has a solution w ith certain initial values and is an infinitesimal function in the vicinity
of an infini te ly remote p oint are de scrib ed . Metho ds of the theory of functions and differential
inequalities in the theory of first-order differential equations are used.