On a heat wave for the nonlinear heat equation: An existence theorem and exact solution

Abstract

The article deals with the nonlinear second order parabolic equation, which is known as heat equation with a source or the generalized porous medium equation. We construct solutions of a special type that describe disturbances propagating over a zero background with a finite velocity. Such solutions are called heat waves. Previously, we considered such problems in cases of one spatial variable, or without a source. In the present work, the obtained results are expanded to the case of two spatial variables. It leads to a transition to the polar coordinate system. The theorem on the existence and uniqueness of a solution, having the form of a heat wave, is proved for one boundary-value problem with special type degeneration in the class of analytical functions. The solution has the form of a multiple power series whose coefficients are determined when solving three-diagonal systems of linear algebraic equations with an infinitely increasing dimension. The convergence of the series is proved by the majorant method. We present an example where the conditions of the theorem are not fulfilled and show that the solution has the form of an immobile heat wave.