Nonlinear differential equation with first order partial derivatives

Abstract

The asymptotic behavior of solutions of a nonlinear differential equation with first-order partial
derivatives solved with respect to one of the derivatives is investigated. Each first-order partial
differential equation under certain conditions has a fundamental system of integrals or an integral
basis. We note that for a general linear partial differential equation of the first order there can
be no nontrivial integral. For a linear homogeneous first-order partial differential equation, where
the coefficients of the equation are given on an unbounded set and have continuous first-order
partial derivatives, with the first coefficient equal to one, an integral basis exists. In this paper,
a nonlinear partial differential equation of the first order, which is solved with respect to one
of the derivatives, is estimated from two sides by first-order partial differential equations. Using
differential inequalities it is proved that a nonlinear differential equation with first-order partial
derivatives solved with respect to one of the derivatives has a solution that tends to zero as one
tends to infinity to one of the independent variables. At present, the theory of partial differential
equations finds its application in various fields of natural science.