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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2022. Vol 42

Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion

Author(s)
Alexander L. Kazakov1,2, Anna A. Lempert1,2

1V. M. Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russian Federation

2 Irkutsk National Research Technical University, Irkutsk, Russian Federation

Abstract
The paper continues a long series of our research and considers a secondorder 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
About the Authors

Alexander L. Kazakov, Doct Sci. (Phys.Mathematics), Prof., V. M. Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664003, Russian Federation, kazakov@icc.ru

Anna A. Lempert, Cand. Sci. (Phys.Math.), V. M. Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664003, Russian Federation, lempert@icc.ru

For citation
Kazakov A. L., Lempert A. A. Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 43–58. https://doi.org/10.26516/1997-7670.2022.42.43
Keywords
parabolic partial differential equations, analytical solution, diffusion wave, existence theorem, exact solution, mathematical modeling.
UDC
517.957
MSC
35K40, 35K57
DOI
https://doi.org/10.26516/1997-7670.2022.42.43
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