Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


In this article, we discuss one problem of nonlinear thermal conductivity with double nonlinearity; an exact analytical solution has been found for it, the analysis of which allows revealing a number of characteristic features of thermal processes in nonlinear media. The following nonlinear effects have been established: the inertial effect, the finite propagation velocity of thermal disturbances, the spatial localization of heat, and the effect of the finite time of the existence of a thermal structure in an absorption medium.

First Page


Last Page



1. Kurdyumov S.P., Zmitrenko N.V. N-and S-modes of compression of the final plasma mass and features of modes with sharpening, PMTF, No.1, 3–23 (1977).

2. Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. and Samarsky A.A. On comparison of solutions of parabolic equations. DAN an SSSR. Vol. 248, No.3, 586–589 (1979).

3. Martinson L.K. Evolution of a heat pulse in a nonlinear medium with volumetric heat absorption. TVT, Vol. 21, No.4, 801–803 (1983).

4. Samarsky A.A., Zmitrenko I.V., Kurdyumov S.P. and Mikhailov A.P. Effect of metastable heat localization in a medium with nonlinear thermal conductivity. DAN USSR, Vol. 223, No.6 (1975).

5. Samarsky A.A., Elenin G.G., Zmitrenko N.V., Kurdyumov S.P. and Mikhailov A.P. Gorenje nonlinear environment in the form of complex structures. DAN USSR, 1977 Vol. 237, No.6, 1330–1333 (1977).

6. Martinson L.K., Pavlov K.B. On the spatial localization of thermal disturbances in the theory of nonlinear thermal conductivity. ZhVM and MF, Vol. 12, No.4, 1048–1053 (1972).

7. Angar Jungel. Cross-Diffusion systems with entropy structure. arXiv: 1710.01623v1 [math.AP] 4 Oct 2017. Proceedings of EQUADIFF, (2017).

8. Aripov M., Abdullaeva Z. On the bottom of the exact solution of a nonlinear problem with absorption or a source. Bulletin of the TATU, No.4, 107–113 (2016).

9. Aripov M., Sadullaeva Sh. Properties of solutions to reaction-diffusion equation with double nonlinearity wih distributed parameters. Log SFU. Ser. Mat and Phys., 6: 2, 157–167 (2013).

10. Tedeev A.F. Conditions for the time global existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations. Siberian Math. J. 45, 155–164 (2004).

11. Wang M., Wei Y. Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J. Math. Anal. Appl. 343, 621–635 (2008).

12. Raimbekov J.R. The Properties of the Solutions for Cauchy Problem of Nonlinear Parabolic Equations in Non-Divergent Form with Density. Journal of Siberian Federal University. Mathematics & Physics, 8(2), 192–200 (2015).

13. Aripov M., Sadullaeva Sh. To properties of solutions to reaction diffusion equation with double nonlinearity with distributed parameters. Jour. of Siberian Fed. Univer. Math. & Phys. 6, 157–167 (2013).

14. Cianci P., Martynenko A.V. and Tedeev A.F. The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source. Nonlinear Analysis: Theory, Methods & Applications A, Vol. 73, No.7, 2310–2323 (2010).

15. Chen C.S., Wang R.Y. Global existence and L1 estimates of solution for doubly degenerate parabolic equation. Acta Mathematica Sinica, Vol. 44, No.6, 1089–1098 (2001).

16. Tsutsumi M. On solutions of some doubly nonlinear degenerate parabolic equations with absorption. Journal of Mathematical Analysis and Applications, Vol. 132, No.1, 187–212 (1988).



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.