Research of compatibility of the redefined system for the multidimensional nonlinear heat equation (special case)

In paper the multidimensional equation of nonlinear heat conductivity is investigated. This equation is presented in the form of the overdetermined system of the differential equations with partial derivatives (the number of the equations are more than number of required functions). It is known that the overdetermined system of the differential equations can be not compatible, at it can not exist any solution. Therefore, for establishment of the fact of existence of solutions and degree of their arbitrariness the analysis of this overdetermined system of the differential equations is carried out. As a result of the conducted research not only sufficient, but also necessary and sufficient conditions of compatibility of the overdetermined system of the differential equations with partial derivatives are received. On the basis of these results with use of the equation of Liouville and the theorem of a necessary and sufficient condition of potentiality of the vector field the approach allowing to construct in some cases exact non-negative solutions of the multidimensional equation of nonlinear heat conductivity with a final velocity of propagation of perturbations is stated. Among the constructed exact decisions are available also such which are not invariant from the point of view of groups of pointed transformations and Lie–Bäcklund's groups. The special attention is paid to the equation with degree-like coefficient of nonlinear heat conductivity. This equation is the quasilinear parabolic equation with implicit degeneration. This equation from the parabolic differential equation of the second order degenerates in the nonlinear evolutionary equation of the first order like Hamilton–Jacobi.