Solutions of the second-order nonlinear parabolic system modeling the diffusion wave motion

A. L. Kazakov ; A. A. Lempert

The Bulletin of Irkutsk State University. Series Mathematics, Tome 42 (2022), pp. 43-58

Résumé

The paper continues a long series of our research and considers a second-order nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical biology, it describes the propagation of pollutants in water and air, as well as population dynamics, including the interaction of two different biological species. We construct solutions that have the type of diffusion (heat) wave propagating over a zero background with a finite velocity. Note that the system degenerates on the line where the perturbed and zero (unperturbed) solutions are continuously joined. A new existence and uniqueness theorem is proved in the class of analytical functions. In this case, the solution has the desired type and is constructed in the form of characteristic series, the convergence of which is proved by the majorant method. We also present two new classes of exact solutions, the construction of which, due to ansatzes of a specific form, reduces to integrating systems of ordinary differential equations that inherit a singularity from the original formulation. The obtained results are expected to be helpful in modeling the evolution of the Baikal biota and the propagation of pollutants in the water of Lake Baikal near settlements.

Détail

Keywords: parabolic partial differential equations, analytical solution, existence theorem, mathematical modeling.
Mots-clés : diffusion wave, exact solution