Exact solutions of the nonlinear heat conduction model

This paper continues a large series of our publications devoted to solutions of the nonlinear heat conduction equation. The solutions are heat waves that propagate over a zero background with a finite velocity. We study the problem on constructing exact solutions of the considered type for the nonlinear heat conduction equation with a source (sink) and determining their properties. A feature of such solutions is that the parabolic type of the equation is degenerate at the front of a heat wave, therefore, properties unusual for parabolic equations appear. We consider two types of solutions. The first one is a simple wave that moves at a constant speed and has the form of a solitary wave (soliton). The second one is a heat wave with an exponential law of front motion. In both cases, the construction is reduced to Cauchy problems for second-order ordinary differential equations (ODEs), which inherit the singularity from the original problem. We construct phase portraits of ODEs and establish the properties of trajectories passing through singular points. Also, we obtain the power series expansions of the required solutions and estimate their convergence radii.