Existence and construction of anisotropic solutions to the multidimensional equation of nonlinear diffusion. I

Siberian Mathematical Journal

Volumn 41, Issue 5, 2000, Pages 940-959

  Rudykh, G A ^a   Semenov E I ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0043024913     PISSN: 00374466     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02674750     Document Type: Article

References (50)

1. Kershner R., "On some properties of generalized solutions to quasilinear degenerate parabolic equations," Acta Math. Hungar., 32, No. 3-4, 301-330 (1978).
2. Antontsev S. N., Localization of Solutions to Degenerate Equations of Continuum Mechanics [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk (1986).
3. Martinson L. K., "Investigation of a mathematical model of the process of nonlinear heat conduction in media with volume absorption," in: Mathematical Simulation. Processes in Nonlinear Media [in Russian] [in Russian], Nauka, Moscow, 1986, pp. 279-309.
4. Kalasbnikov A. S., "Some questions of the qualitative theory of nonlinear degenerate parabolic equations of the second order," Uspekhi Mat. Nauk, 42, No. 2, 135-176 (1987).
5. Galaktionov V. A., Dorodnitsyn V. A., Elenin G. G., Kurdyumov S. P., and Samarskiǐ A. A., "A quasilinear heat equation with a source: blow-up, localization, symmetry, exact solutions, asymptotics, and structures," in: Contemporary Problems of Mathematics. Modern Achievements [in Russian] VINITI, Moscow, 1987, 28, pp. 95-205. (Itogi Nauki i Tekhniki.)
6. C. Samarskiǐ A. A., Galaktionov V. A., Kurdyumov S. P., and Mikhaǐlov A. P., Blowups in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).
7. Aronson D. G., "Regularity of flows in porous media: a survey," in: Nonlinear Diffusion Equations and Their Equilibrium States, Springer-Verlag, New York, 1988, 1, pp. 35-49.
8. Galaktionov V. A. and Posashkov S. A., Unbounded Exact Solution of the Nonlinear Heat Equation with Source [Preprint, No. 42], Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow(1988).
9. Galaktionov V. A. and Posashkov S. A., "On new exact solutions to parabolic equations with quadratic nonlinearities," Zh. Vychisl. Mat. i Mat. Fiz., 29, No. 4, 497-500 (1989).
10. Galaktionov V. A. and Posashkov S. A., Examples of Asymmetric Complete Cooling off and Blowups for Quasilinear Heat Equations [in Russian] [Preprint, No. 21], Inst. Prikl. Mat. Ross. Akad. Nauk, Moscow (1994).
11. Galaktionov V. A. and Posashkov S. A., "Exact solutions and invariant spaces for nonlinear equations of gradient diffusion," Zh. Vychisl. Mat. i Mat. Fiz., 34, No. 3, 373-383 (1994).
12. Galaktionov V. A. and Posashkov S. A., and Svishcherskiǐ S. P., "Generalized separation of variables for differential equations with polynomial nonlinearities," Differentsial'nye Uravneniya, 31, No. 2, 253-261 (1995).
13. Galaktionov V. A., "On new exact blow-up solutions for nonlinear heat conduction equations with source and applications," Differential Integral Equations, 3, No. 5, 863-874 (1990).
14. Galaktionov V. A., Invariant Subspaccs and New Explicit Solutions to Evolution Equations with Quadratic Nonlinearities, School of Mathematics, Univ. Bristol, Bristol (1991). (Report No. AM-91-11.)
15. Galaktionov V. A., "Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities," Proc. Roy. Soc. Edinburgh Sect. A, 125, No. 2, 225-246 (1995).
16. King J. R., "Exact multidimensional solutions to some nonlinear diffusion equations," Quart. J. Mech. Appl. Math., 46, No. 3, 419-436 (1993).
17. Pukhnachëv V. V., "Equivalence transformations and hidden symmetry of evolution equations," Dokl. Akad. Nauk SSSR, 294, No. 3, 535-538 (1989).
18. Pukhnachëv V. V., "Reciprocal transformations of radial equations of nonlinear heat conduction," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 213, 151-163 (1994).
19. Pukhnachev V. V., "Multidimensional exact solutions of a nonlinear diffusion equation," Prikl. Mekh. Tekhn. Fiz., 36, No. 2, 23-31 (1995).
20. Meirmanov A. M., Pukhnachev V. V., and Shmarev S. I., Evolution Equations and Lagrangian Coordinates, Walter de Gruyter, Berlin and New York (1997).
21. Kosygina E. R., "On anisotropic exact solutions of a multidimensional equation of nonstationary filtration," Zh. Vychisl. Mat. i Mat. Fiz., 35, No. 2, 241-259 (1995).
22. Olver P. J., Symmetry and Explicit Solutions of Partial Differential Equations [Preprint/Univ. of Minnesota] (1991).
23. Olver P. J., "Direct reduction and differential constrains," Proc. Roy. Soc. London, 444, No. 1922, 509-523 (1994).
24. Sidorov A. F., "Analytic representations of solutions of nonlinear parabolic equations of nonstationary filtration type," Dokl. Akad. Nauk SSSR, 280, No. 1, 47-51 (1985).
25. Titov S. S., "On the propagation of a nonlinear-diffusion front," Prikl. Mekh. i Tekhn. Fiz., 37, No. 4, 113-118 (1996).
26. Bertsch M., Kersner R., and Peletier L. A., "Positivity versus localization in degenerate diffusion equations," Nonlinear Anal., Theory Methods Appl., 9, No. 10, 987-1008 (1985).
27. Rudykh G. A. and Semënov È. I., Commutative Representations and Bäcklund Transformations for Nonlinear Evolution Equations with One Space Variable [Preprint, No. 7] [in Russian], Irkutsk. Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk (1990).
28. Rudykh G. A. and Semenov E. I., "Application of Liouville's equation to construction of special exact solutions for the quasilinear heat equation," IMACS Ann. Comput. Appl. Math., 8, 193-196 (1990).
29. Rudykh G. A. and Semenov E. I., On a Certain Approach to Constructing Particular Exact Solutions to the Quasilinear Heat Equation in N Spatial Variables [in Russian] [Preprint, No. 6] Irkutsk. Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk (1991).
30. Rudykh G. A. and Semenov E. I., "Construction of exact solutions to a multidimensional quasilinear heat equation," Zh. Vychisl. Mat. i Mat. Fiz., 33, No. 8, 1228-1239 (1993).
31. Rudykh G. A. and Semenov E. I., "New exact solutions of the one-dimensional nonlinear diffusion equation," Sibirsk. Mat. Zh., 38, No. 5, 1130-1139 (1997).
32. Rudykh G. A. and Semenov E. I., "On new exact solutions of the one-dimensional nonlinear diffusion equation with a source (sink)," Zh. Vychisl. Mat. i Mat. Fiz., 38, No. 6, 971-977 (1998).
33. Rudykh G. A. and Semenov E. I., "Exact nonnegative solutions to the multidimensional nonlinear diffusion equation," Sibirsk. Mat. Zh., 39, No. 5, 1129-1138 (1998).
34. Rudykh G. A. and Semenov E. I., "Nonselfsimilar solutions of the multidimensional nonlinear diffusion equation," Mat. Zametki, 67, No. 2, 250-256 (2000).
35. Rudykh G. A., "Exact selfsimilar solutions of the multidimensional nonlinear diffusion equation," Dokl. Akad. Nauk, 358, No. 3, 323-324 (1998).
36. Ovsyannikov L. V., Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
37. Ibragimov N. Kh., Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983).
38. Oleǐnik O. A., Kalashnikov A. S., and Chzhou Yuǐ-Lin', "The Cauchy problem and boundary value problems for nonstationary filtration type equations," Izv. Akad. Nauk SSSR Ser. Mat., 22, No. 5, 007-704 (1958).
39. Kalashnikov A. S., "On formation of singularities in solutions to the nonstationary filtration equation," Zh. Vychisl. Mat. i Mat. Fiz., 7, No. 2, 440-443 (1967).
40. Kalashnikov A. S., "On nonstationary filtration type equations with infinite propagation speed of perturbations," Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 45-49 (1972).
41. Maslov V. P., Danilov V. G., and Volosov K. A., Mathematical Modeling of Heat and Mass Transfer Processes [in Russian], Nauka, Moscow (1987).
42. Hirota R., "Direct methods in soliton theory," in: Solitons [Russian translation], Mir, Moscow, 1983, pp. 175-192.
43. Rozhdestvenskiǐ B. L. and Yanenko N. N., Systems of Quasilinear Equations and Their Applications to Gas Dynamics [in Russian], Nauka, Moscow (1978).
44. Titov S. S., "The method of finite-dimensional rings for solving nonlinear equations of mathematical physics," Aèrodinamika, No. 11, 104-110 (1988).
45. Pokhozhaev S. I., "On a problem by L. V. Ovsyannikov," Prikl. Mekh. i Tekhn. Fiz., No. 2, 5-10 (1989).
46. Galaktionov V. A. and Posashkov S. A., and Svishcherskiǐ S. P., On Invariant Sets and Exact Solutions to Nonlinear Evolution Equations with Quadratic Nonlinearities [Preprint, No. 22], Inst. Prikl. Mat. Ros. Akad. Nauk, Moscow(1994).
47. Horn R. and Johnson Ch., Matrix Analysis [Russian translation], Mir, Moscow (1989).
48. Akhiezer N. I., Elements of the Theory of Elliptic Functions [in Russian], Gostekhizdat, Moscow (1948).
49. Hartman P., Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).
50. Kato T., Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).