Explicit estimate for blow-up solutions of nonlinear parabolic systems of non-divergence form with variable density

In this paper, studied the blow-up properties of solutions of nonlinear parabolic systems of non-divergent equations with a variable density. In solving the Cauchy problems for the equations of partial differential equations by the difference method important issue is the choice of the order of approximation with respect to the spatial coordinates and the choose of a function as an initial approximation, as well as the search of an efficient algorithm for solving the systems of difference equations. We construct the difference scheme of high accuracy for nonlinear system of non-divergence form.