## List of issues > Series «Mathematics». 2018. Vol. 24

##
On Analytic Solutions of the Problem of Heat Wave Front Movement for the Nonlinear Heat Equation with Source

We continue our investigation of the special boundary-value problems for the nonlinear parabolic heat equation (the porous medium equation) in the article. In the case of a power-law dependence of the heat conductivity coefficient on temperature the equation is used for describing high-temperature processes, filtration of gas through porous media, migration of biological populations, etc. Moreover, the equation has specific nonlinear properties, which may be interesting from the point of view of physics as well as mathematics. For example, it is well known, that the described disturbances may have a finite velocity of propagation. The heat waves (waves of filtration) compose an important class of solutions to the equation under consideration. Geometrically, these solutions are constructed from two integral surfaces, which are continuously connected on a curve - heat wave front. We consider a boundary-value problem, which has such solutions. The research is carried out in the class of analytic functions by the characteristic series method. This method was suggested by R. Courant and then it was adapted for nonlinear parabolic equations in A.F. Sidorov’s scientific school. We have already researched similar problems in case of closed front without source. For each problem we constructed the solution in form of characteristic series and proved the exist theorem, which guaranteed the convergence. The paper deals with the flat-symmetrical problem with given front and source. The theorem of existence of the analytic solution(heat wave’s nonnegative part) was proved and the solution in form of the power series was constructed. Also we considered an interesting case in which the source is a power function (such cases are common in applications). It was shown that the original problem may be reduced to the Cauchy problem for nonlinear ordinary differential equation of the second order.

**About the Authors**

Alexandr L. Kazakov, Dr. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation, e-mail: kazakov@icc.ru

Pavel A. Kuznetsov, Cand. Sci. (Phys.–Math.), Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation, e-mail: pav_ku@mail.ru

**For citation:**

*The Bulletin of Irkutsk State University. Series Mathematics,*2018, vol. 24, pp. 37-50. (in Russian) https://doi.org/10.26516/1997-7670.2018.24.37

1. Zel’dovich Ya.B., Raizer Yu.P. Physics of Shock Waves and High Temperature Hydrodynamics Phenomena. New York, Dover Publications, 2002, 944 p.

2. Kazakov A.L., Kuznetsov P.A. On One Boundary Value Problem for a Nonlinear Heat Equation in the Case of Two Space Variables. Journal of Applied and Industrial Mathematics, 2014, vol. 8, no. 2, pp. 1-11.https://doi.org/10.1134/S1990478914020094

3. Kazakov A.L., Kuznetsov P.A., Lempert A.A. On Construction of Heat Wave for Nonlinear Heat Equation in Symmetrical Case. The Bulletin of Irkutsk State University. Series Mathematics, 2015, vol. 11, pp. 39-53. (in Russian)

4. Kazakov A.L., Kuznetsov P.A., Spevak L.F. On a Degenerate Boundary Value Problem for the Porous Medium Equation in Spherical Coordinates. Trudy Instituta matematiki i mehaniki UrO RAN, 2014, vol. 20, no. 1, pp. 119-129. (inRussian)

5. Kazakov A.L., Lempert A.A. Analytical and Numerical Investigation of a Nonlinear Filtration Boundary-Value Problem with Degeneration. Vychislitel’nye tehnologii, 2012, vol. 17, no. 1, pp. 57-68. (in Russian)

6. Kazakov A.L., Lempert A.A. Existence and Uniqueness of the Solution of the Boundary-Value Problem for a Parabolic Equation of Unsteady Filtration. Journal of Applied Mechanics and Technical Physics, 2013, vol. 54, no. 2, pp. 251-258.https://doi.org/10.1134/S0021894413020107

7. Kazakov A.L., Spevak L.F. Boundary Elements Method and Power Series Method for One-dimensional Nonlinear Filtration Problems. The Bulletin of Irkutsk State University. Series Mathematics, 2012, vol. 5, no. 2, pp. 2-17. (in Russian)

8. Kovalev V.A., Kurkina E.S., Kuretova E.D. Thermal self-focusing during solar flares. Plasma Physics Reports, 2017, vol. 43, no. 5, pp. 583-587. https://doi.org/10.1134/S1063780X17050063

9. Kudryashov N.A., Chmykhov M.A. Approximate Solutions to One-Dimensional Nonlinear Heat Conduction Problems with a Given Flux. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 1, pp. 107-117.https://doi.org/10.1134/S0965542507010113

10. Courant R., Hilbert D. Methods of Mathematical Physics. Vol. II: Partial Differential Equations. New York, Interscience, 2008.878 p.

11. Rudykh G.A., Semenov E.I. Construction of Exact Solutions of One-Dimensional Nonlinear Diffusion Method of Linear Invariant Subspaces. The Bulletin of Irkutsk State University. Series Mathematics, 2013, vol. 6, no. 4, pp. 69-84. (in Russian)

12. Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in quasilinear parabolic equations. Walter de Gruyte Berlin, NY, 1995. 534 p.

13. Sidorov A.F. Selected Works: Mathematics. Mechanics. Moscow, Fizmatlit Publ., 2001. 576 p. (in Russian)

14. Dahlberg B.E., Kenig C.E. Weak Solutions of the Porous Medium Equation in a Cylinder. Trans. Amer. Math. Soc., 1993, vol. 336, no. 2, pp. 701-709. https://doi.org/10.1090/S0002-9947-1993-1085940-6

15. Filimonov M.Yu., Korzunin L.G., Sidorov A.F. Approximate methods for solving nonlinear initial boundary-value problems based on special construction of series Rus. J. Numer. Anal. Math. Modelling, 1993, vol. 8, no. 2, pp. 101-125.https://doi.org/10.1515/rnam.1993.8.2.101

16. Murray J. Mathematical Biology: I. An Introduction, Third Edition. Interdisciplinary Applied Mathematics. Vol. 17. New York, Springer, 2002, 551 p.

17. Vazquez J.L. The Porous Medium Equation: Mathematical Theory. Oxford, Clarendon Press, 2007, 648 p.