Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Abstract

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.