On periodic solutions of a nonlinear reaction-diffusion system

A. A. Kosov ; E. I. Semenov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 26 (2018), pp. 35-46

Résumé

We consider a system of three parabolic partial differential equations of a special reaction-diffusion type. In this system, the terms that describe diffusion are identical and linear with constants coefficients, whereas reactions are described by homogenous polynomials of degree 3 that depend on three parameters. The desired functions are considered to be dependent on time and an arbitrary number of spatial variables (a multi-dimensional case). It has been shown that the reaction-diffusion system under study has a whole family of exact solutions that can be expressed via a product of the solution to the Helmholz equations and the solution to a system of ordinary differential equations with homogenous polynomials, taken from the original system, in the right-hand side. We give the two first integrals and construct a general solution to the system of three ordinary differential equations, which is represented by the Jacobi elliptic functions. It has been revealed that all particular solutions derived from the general solution to the system of ordinary differential equations are periodic functions of time with periods depending on the choice of initial conditions. Additionally, it has been shown that this system of ordinary differential equations has blow-up on time solutions that exist only on a finite time interval. The corresponding values of the first integrals and initial data are found through the equality conditions. A special attention is paid to a class of radially symmetric with respect to spatial variables solutions. In this case, the Helmholz equation degenerates into an non-autonomous linear second-order ordinary differential equation, which general solution is found in terms of the power functions and the Bessel functions. In a particular case of three spatial variables the general solution is expressed using trigonometric or hyperbolic functions.

Détail

Keywords: reduction to a system of ODEs, periodic solutions, Jacobi elliptic functions.
Mots-clés : reaction-diffusion system, exact solutions

Bibliographie

[1] Axiezer N. I., Elements of the theory of elliptic functions, OGIZ, Gostexizdat Publ., M., 1948, 292 pp. (in Russian)

[2] Kapcov O. V., Methods for the integration of partial differential equations, Fizmatlit Publ., M., 2009, 184 pp. (in Russian)

[3] Kosov A. A., Semenov E. I., “Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms”, Siberian Math. J., 58:4 (2001), 619–632 | DOI | MR | MR | Zbl

[4] Kosov A. A., Semenov E. I., “First integrals and periodic solutions of a system with power nonlinearities”, Journal of Applied and Industrial Mathematics, 12:1 (2018), 70–83 | DOI | MR | Zbl

[5] Kosov A. A., Semenov E. I., “On analytic periodic solutions of nonlinear differential equations with delay (lead)”, Izvestiya vysshih uchebnyh zavedenij. Matematika, 2018, no. 10, 34–42 (in Russian) | Zbl

[6] Polyanin A. D., Zaitsev V. F., Handbook of Nonlinear Partial Differential Equations, Second Edition, Updated, Revised and Extended, Chapman $\$ Hall/CRC Press, Boca Raton–London–New York, 2012, 1912 pp. | MR

[7] Polyanin A. D., Zajcev V. F., Nonlinear equations of mathematical physics, Tutorial. In 2 parts, v. 1, Yurait Publ., M., 2017, 324 pp. (in Russian)

[8] Polyanin A. D., Zajcev V. F., Nonlinear equations of mathematical physics, Tutorial. In 2 parts, v. 2, Yurajt Publ., M., 2017, 370 pp. (in Russian)

[9] Slin'ko M.G., Zelenyak T. I., Akramov T. A., Lavrent'ev M. M.-jr., Scepelev V. S., “Nonlinear dynamics of catalytic reactions and processes (review)”, Mat. modelirovanie, 9:12 (1997), 87–100 (in Russian) | MR

[10] Shmidt A. V., “Analysis of reaction-diffusion systems by the method of linear defining equations”, Zhurn. vychisl. matem. i matem. fiziki, 47:2 (2007), 256–268 (in Russian) | MR

[11] Cherniha R., King J. R., “Non-linear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansatze and exact solutions”, J. Math. Anal. Appl., 308 (2005), 11–35 | DOI | MR | Zbl

[12] Galaktionov V. A., Svirshchevskii S. R., Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, Boca Raton, London; Taylor $\$ Francis Group, New York, 2007 | MR | Zbl

[13] Meirmanov A. M., Pukhnachov V. V., Shmarev S. I., Evolution Equations and Lagrangian Coordinates, Walter de Gruyter, Berlin–New York, 1997 | MR

[14] Polyanin A. D., Kutepov A. M., Vyazmin A. V., Kazenin D. A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor $\$ Francis, London–NY, 2002, 387 pp.

[15] Nefedov N. N., Nikulin E. I., “Existence and Stability of Periodic Solutions for Reaction-Diffusion Equations in the Two-Dimensional Case”, Modeling and Analysis of Information Systems, 23:3 (2016), 342–348 | DOI | MR

[16] Vazquez J. L., The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2007 | MR