Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, 664033 Irkutsk, Russia
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Symmetry 2020, 12(6), 999; https://doi.org/10.3390/sym12060999
Received: 26 May 2020
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Revised: 6 June 2020
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Accepted: 6 June 2020
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Published: 11 June 2020
(This article belongs to the Special Issue Solvability of Nonlinear Equations with Parameters: Branching, Regularization, Group Symmetry and Solutions Blow-Up)
The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.
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Keywords:
reaction-diffusion system; diffusion wave; existence theorem; analytical solution; power series; majorant method; exact solution
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MDPI and ACS Style
Kazakov, A.; Kuznetsov, P.; Lempert, A. Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. Symmetry 2020, 12, 999. https://doi.org/10.3390/sym12060999
AMA Style
Kazakov A, Kuznetsov P, Lempert A. Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. Symmetry. 2020; 12(6):999. https://doi.org/10.3390/sym12060999
Chicago/Turabian StyleKazakov, Alexander, Pavel Kuznetsov, and Anna Lempert. 2020. "Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type" Symmetry 12, no. 6: 999. https://doi.org/10.3390/sym12060999
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