Research of compatibility of the redefined system for the multidimensional nonlinear heat equation

G. A. Rudykh   ; È. I. Semenov

Matematičeskie zametki SVFU, Tome 25 (2018) no. 1, pp. 50-62

Résumé

We study the multidimensional parabolic second-order equation with the implicit degeneration and the finite velocity of propagation of perturbations. This equation is given in the form of an overdetermined system of the differential equations with partial derivatives (the number of the equations exceeds the number of the required functions). It is known that an overdetermined system of the differential equations may not be compatible as well as may not have any solutions. Therefore, in order to determine the existence of the solutions and the degree of their arbitrariness the analysis of this overdetermined system is carried out. As a result of the research, the sufficient and the necessary and sufficient compatibility conditions for the overdetermined system of the differential equations with partial derivatives are received. On the basis of these results with the use of the equation of Liouville and the theorem of the potential operators, the exact non-negative solutions of the multidimensional nonlinear heat equation with the finite velocity of propagation of perturbations are constructed. In addition, the new exact non-negative solutions of the nonlinear evolution of Hamilton-Jacobi equations are obtained; the solutions of the nonlinear heat equation and the solutions of Riemann wave equation are also found. Some solutions are not invariant from the point of view of the groups of the pointed transformations and Lie-Bäcklund's groups. Finally, the transformations of Bäcklund linking the solutions of the multidimensional nonlinear heat equation with the related nonlinear evolution equations are obtained.

Détail

Keywords: multidimensional nonlinear heat equation, nonlinear evolution equations, finite velocity of propagation of perturbation, Bäcklund transformation.
Mots-clés : exact nonnegative solutions

Bibliographie

[1] Kaplan W., “Some methods for analysis of the flow in phase space”, Proc. of the symposium on nonlinear circuit analysis (April 23-24), Polytechnic Institute of Brooklyn, New York, 1953, 99–106 | MR

[2] Steeb W. H., “Generalized Liouville equation, entropy and dynamic systems containing limit cycles”, Physica, 95:1 (1979), 181–190 | DOI | MR

[3] Vainberg M. M., Variational Methods for Nonlinear Operators Study, Gostekhteorizdat, Moscow, 1956

[4] Berger M. S., Perspectives in nonlinearity, Benjamin Publishers, New York; Amsterdam, 1968 | Zbl

[5] Kalashnikov A. S., “Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations”, Russ. Math. Surv., 42:2 (1987), 135–176 | MR | Zbl

[6] Galaktionov V. A., Dorodnitsyn V. A., Elenin G. G., Kurdyumov S. P., Samarskii A. A., “A quasilinear equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures”, J. Sov. Math., 41 (1988), 1222–1292 | DOI | MR

[7] Samarskij A. A., Galaktionov V. A., Kurdyumov S. P., Mikhajlov A. P., Peaking Models in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, 1987 | MR

[8] Ovsyannikov L. V., Group Analysis of Differential Equations, Nauka, Moscow, 1978

[9] Ibragimov N. H., Transformation Groups in Mathematical Physics, Nauka, Moscow, 1983

[10] Antontsev S. N., Localization of Solutions to Degenerate Equations of Continuum Mechanics, Lavrentiev Hydrodyn. Inst., Novosibirsk, 1986

[11] Aronson D. G., The porous medium equation. Some problems in nonlinear diffusion, Lecture Notes in Math., 1224, Springer-Verl., Berlin, 1986 | MR

[12] Aronson D. G., “Regularity of flows in porous medium: a survey”, Nonlinear diffusion equations and their equilibrium states, 1:1 (1988), 35– 49, Springer-Verl., New York | DOI | MR

[13] Olejnik O. A., Kalashnikov A. S., Zhou Yu-lin, “The Cauchy problem and boundary problems for equations of the type of non-stationary filtration”, Izv. Akad. Nauk SSSR, 22:5 (1958), 667–704 | Zbl

[14] Kalashnikov A. S., “The occurrence of singularities in solutions of the non-steady seepage equation”, Zh. Vychisl. Mat. Mat. Fiz., 7:2 (1967), 440–443

[15] Kalashnikov A. S., “On equations of the type of non-stationary filtration with infinite velocity of perturbation propagation”, Vestn. Moskov. Univ., Ser. I, 1972, no. 6, 45–49

[16] Yanenko N. N.,, “Compatibility theory and integration methods for systems of partial differential equations”, Tr. Vsesoyuz. Mat. S'ezda, v. 2, Nauka, Leningrad, 1964, 613–621

[17] Rozhdestvenskii B. L., Systems of Quasilinear Equations, Nauka, Moscow, 1978

[18] Sidorov A. F., Metod Differentsialnyh Svyazey i Ego Prilozheniya v Gazovoy Dinamike, Nauka, Novosibirsk, 1984

[19] Shapeev V. P., Metod Differentsialnyh Svyazey i Ego Prilozheniya k Uravneniyam Mekhaniki Sploshnoy Sredy, Diss. Dokt. Fiz.-Mat. Nauk, Novosibirsk, 1987

[20] Kaptsov O. V., “Linear determining equations for differential constraints”, Sb. Math., 189:12 (1998), 1839–1854 | DOI | DOI | MR | Zbl

[21] Kaptsov O. V., Integration Methods for Partial Differential Equations, Fizmatlit, Moscow, 2009

[22] Rudykh G. A., Semenov E. I., “Research of compatibility of the redefined system for the multidimensional nonlinear heat equation (special case)”, Izv. Irkutsk. Gos. Univ., Ser. Mat., 18 (2016), 93–109 | Zbl

[23] Vainberg M. M., Functional Analysis, Prosveshchenie, Moscow, 1979

[24] Newell A. C., Solitons in Mathematics and Physics, SIAM, 1989 | MR | MR

[25] Bogoyavlenskii O. I., Overturning Solitons, Nauka, Moscow, 1991 | MR

[26] Holden H., “The Burgers equation with a noisy force and the stochastic heat equation”, Commun. Part. Diff. Equ., 19:1-2 (1994), 119–141 | DOI | MR | Zbl

[27] Bitsadze A. V., “Exact solutions of some classes of nonlinear partial differential equations”, Differ. Equations, 17:10 (1981), 1774–1778 | MR

[28] Bitsadze A. V., Some Classes of Partial Differential Equations, Nauka, Moscow, 1981