Research of compatibility of the redefined system for the multidimensional nonlinear heat equation (special case)

G. A. Rudykh   ; E. I. Semenov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 18 (2016), pp. 93-109

Résumé

In paper the multidimensional equation of nonlinear heat conductivity is investigated. This equation is presented in the form of the overdetermined system of the differential equations with partial derivatives (the number of the equations are more than number of required functions). It is known that the overdetermined system of the differential equations can be not compatible, at it can not exist any solution. Therefore, for establishment of the fact of existence of solutions and degree of their arbitrariness the analysis of this overdetermined system of the differential equations is carried out. As a result of the conducted research not only sufficient, but also necessary and sufficient conditions of compatibility of the overdetermined system of the differential equations with partial derivatives are received. On the basis of these results with use of the equation of Liouville and the theorem of a necessary and sufficient condition of potentiality of the vector field the approach allowing to construct in some cases exact non-negative solutions of the multidimensional equation of nonlinear heat conductivity with a final velocity of propagation of perturbations is stated. Among the constructed exact decisions are available also such which are not invariant from the point of view of groups of pointed transformations and Lie–Bäcklund's groups. The special attention is paid to the equation with degree-like coefficient of nonlinear heat conductivity. This equation is the quasilinear parabolic equation with implicit degeneration. This equation from the parabolic differential equation of the second order degenerates in the nonlinear evolutionary equation of the first order like Hamilton–Jacobi.

Détail

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

Bibliographie

[1] Antontsev S. N., Localization of solutions of degenerate equations of continuum mechanics, Institut gidrodinamiki SO AN SSSR, Novosibirsk, 1986, 108 pp. (in Russian)

[2] Bellman R., Introduction to the theory of matrices, Nauka, M., 1976 (in Russian)

[3] Vajnberg M. M., Variational methods for the study of nonlinear operators, Gostekhizdat, M., 1956 (in Russian)

[4] Vladimirov V. S., Equations of mathematical physics, Nauka, M., 1971 (in Russian)

[5] Galaktionov V. A., Dorodnicyn V. A., Elenin G. G., Kurdyumov S. P., Samarskij A. A., “The quasilinear heat equation: peaking, localization, symmetry, exact solutions, asymptotic behavior, structures”, Sovrem. probl. matem. Novejshie dostizheniya. Itogi nauki i tekhniki, 28, VINITI AN SSSR, M., 1987, 95–205 (in Russian)

[6] Gantmaher F. R., The theory of matrices, Nauka, M., 1966 (in Russian)

[7] Ibragimov N. H., Groups of transformations in mathematical physics, Nauka, M., 1983 (in Russian)

[8] Kalashnikov A. S., “On the occurrence of singularities in the solutions of the equation unsteady filtration”, Zhurn. vychis. matem. i matem. fiziki, 7:2 (1967), 440–443 (in Russian)

[9] Kalashnikov A. S., “On the equations of unsteady filtration type with infinite perturbation propagation velocity”, Vestn. MGU. Ser. mat. mekh., 1972, no. 6, 45–49 (in Russian)

[10] Kalashnikov A. S., “Some questions in the qualitative theory of nonlinear degenerate parabolic equations of second order”, UMN, 42:2 (1987), 135–176 (in Russian) | Zbl

[11] Kaptsov O. V., “Linear determining equations for differential constraints”, Matem. sbornik, 189:12 (1998), 103–118 (in Russian) | DOI | Zbl

[12] Kaptsov O. V., Methods of integration of partial differential equations, Fizmatlit, M., 2009, 184 pp. (in Russian)

[13] Martinson L. K., “A study of mathematical model of the non-linear transfer thermal conductivity with volume absorption media”, Matematicheskoe modelirovanie. Processy v nelinejnyh sredah, Nauka, M., 1986, 279–309 (in Russian)

[14] Ovsyannikov L. V., Group analysis of differential equations, Nauka, M., 1978 (in Russian)

[15] Olejnik O. A., Kalashnikov A. S., Chzhou-Yuj-Lin', “Cauchy problem and boundary value problems for equations of unsteady filtration”, Izv. AN SSSR. Ser. mat., 22:5 (1958), 667–704 (in Russian) | Zbl

[16] Rozhdestvenskij B. L., Yanenko N. N., Systems of quasilinear equations, Nauka, M., 1978 (in Russian)

[17] Rudykh G. A., Semenov E. I., An approach of constructing exact solutions of partial quasilinear heat equation with $N$-spatial variables, Preprint No 6, IrVC SO AN SSSR, Irkutsk, 1991, 21 pp. (in Russian)

[18] Rudykh G. A., Semenov E. I., “Construction of exact solutions of the multidimensional quasilinear heat equation”, Computational Mathematics and Mathematical Physics, 33:8 (1993), 1087–1097 | MR | Zbl

[19] Samarskij A. A., Galaktionov V. A., Kurdyumov S. P., Mihajlov A. P., Modes with peaking in problems for quasi-linear parabolic equations, Nauka, M., 1987, 480 pp. (in Russian)

[20] Sidorov A. F., Shapeev V. P., Yanenko N. N., The method of differential constraints and its applications in gas dynamics, Nauka, Novosibirsk, 1984 (in Russian)

[21] Horn R., Dzhonson Ch., Matrix analysis, Mir, M., 1989 (in Russian)

[22] Shapeev V. P., The method of differential constraints and its application to the equations Continuum Mechanics, Diss. dokt. fiz.-mat. nauk, Novosibirsk, 1987 (in Russian)

[23] Yanenko N. N., “The theory of compatibility and integration methods of nonlinear systems PDEs”, Trudy IV Vsesoyuznogo matematicheskogo s"ezda, v. 2, Nauka, L., 1964, 613–621 (in Russian) | Zbl

[24] D. G. Aronson, “The porous medium equation”, Some problems in nonlinear diffusion, Lecture Notes in Math., 1224, Springer Verlag, 1986 | MR

[25] D. G. Aronson, “Regularity of flows in porous medium: a survey”, Nonlinear Diffusion Equations and Their Equilibrium States, 1:1 (1988), 35–49 | DOI | MR

[26] M. S. Berger, Perspectives in nonlinearity, N.-Y.–Amsterdam, 1968

[27] W. Kaplan, “Some methods for analysis of the flow in phase space”, Proc. of the symposium on nonlinear circuit analysis, N.Y., 1953, 99–106 | MR | Zbl

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

[29] G. A. Rudykh, E. I. Semenov, “Application of Liouville's equation to construction of special exact solutions for the quasilinear heat equation”, IMACS Ann. Comput. and Appl. Math., 8 (1990), 193–196 | MR

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

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