Local Existence of the Solution to the Initial-Boundary Value Problem in Nonlinear Thermodiffusion in Micropolar Medium

Abstract

We prove a theorem about local existence (in time) of the solution to the first initial-boundary value problem for a nonlinear hyperbolic-parabolic system of eight coupled partial differential equations of second order describing the process of thermodiffusion in a three-dimensional micropolar medium. At first, we prove existence, uniqueness and regularity of the solution to this problem for the associated linearized system by using the Faedo-Galerkin method and semi-group theory. Next, we prove (basing on this theorem) an energy estimate for the solution to the linearized system by applying the method of Sobolev spaces. At last, by using the Banach fixed point theorem we prove that the solution of our nonlinear problem exists and is unique.