On the Analytical Construction of a Heat Wave for the Nonlinear Heat Equation With a Source in Polar Coordinates

Abstract

The paper is devoted to the study of a nonlinear second-order parabolic equation, which in the literature is called the heat equation with a source or the generalized porous medium equation. We construct specialized solutions that describe disturbances propagating over the zero background at a finite velocity (heat waves). Previously, we studied such problems without a source. In this paper, we extend the known results to a more general case. The theorem of the existence and uniqueness of a solution having the form of a heat wave in polar coordinates is proved. A heat wave is constructed in the form of a convergent multiple power series, the c oefficients of which are determined when solving systems of linear algebraic equations. We give an example where the conditions of the theorem are not satisfied. It shows that the solution, in this case, has the form of a stable heat wave.