On Boundary Value Problem with Degeneration for Nonlinear Porous Medium Equation with Boundary Conditions on the Closed Surface

The paper deals with the special initial boundary value problem for nonlinear heat equation in $\mathbb{R}^3$ in case of power dependence of heat-conduction coefficient on temperature. In English scientific publications this equation is usually called the porous medium equation. Nonlinear heat equation is used for mathematical modeling of filtration of polytropic gas in the porous medium, blood flow in small blood vessels, processes of the propagation of emissions of negative buoyancy in ecology, processes of growth and migration of biological populations and other. The unknown function is equal to zero in initial time and heating mode is given on the closed sufficiently smooth surface in considered problem. The transition to the spherical coordinate system is performed. The theorem of existence and uniqueness of analytic solution of the problem is proved. The solution has type of heat wave which has finite velocity of propagation. The procedure of construction of the solution in form of the power series is proposed. The coefficients of the series are founded from systems of linear algebraic equations. Since the power series coefficients is constructed explicitly, this makes it possible to use the solution for verification of numerical calculations.