Existence and Uniqueness of a Classical Solution of Fourier’s First Problem for Nonlinear Parabolic-elliptic Systems

This paper deals with the existence and uniqueness of the classical solution of Fourier’s first problem for a wide class of systems of two weakly coupled quasi-linear second order partial differential functional equations. One equation is of the parabolic type (the degenerated parabolic equation) and the other of the elliptic type (the elliptic equation with a parameter). The functional dependence is of the Volterra type. The differential functional problem is considered in the one-dimensional case. A suitable theorem is formulated and proved. The proof is based on some monotone iterative method with use of Green’s function and basic theorems of integral calculus. It is a new technique of solving of the specific mixed systems considered. Examples of physical applications are given.