Methods of constructing approximate self-similar solutions of nonlinear

V. A. Galaktionov   ; A. A. Samarskii

Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 291-321

Résumé

A rather general approach is presented to the investigation of the asymptotic behavior of solutions to boundary value problems for quasilinear parabolic equations $$ \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\biggl(k(u)\frac{\partial u}{\partial x}\biggr) $$ with arbitrary coefficients $k(u)>0$, $u>0$, and arbitrary boundary regimes $u(t,0)=\psi(t)$ (the problem is considered in the half space $x \in(0,+\infty)$). The investigation is carried out by constructing so-called approximate self-similar solutions which do not satisfy the equation but to which the solution of the problem converges asymptotically in special norms. In this paper the case $[k(u)/k'(u)]'-1/\sigma$ as $u\to+\infty$, $\sigma=\operatorname{const}t>0$, is considered. Bibliography: 61 titles.

Bibliographie

[1] Zeldovich Ya. B., Raizer Yu. P., Fizika udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavlenii, Nauka, M., 1966

[2] Zmitrenko N. V., Kurdyumov S. P., “$N$- i $S$-rezhimy szhatiya konechnoi massy plazmy i osobennosti rezhimov s obostreniem”, PMTF, 1977, no. 1, 3–23

[3] Kurdyumov S. P., Lokalizatsiya diffuzionnykh protsessov i vozniknovenie struktur pri razvitii v dissipativnoi srede rezhimov s obostreniem, Diss. na soisk. uchenoi stepeni dokt. fiz.-mat. nauk, Inst. prikl. matem. im. M. V. Keldysha AN SSSR, M., 1979

[4] Zeldovich Ya. B., Kompaneets A. S., “K teorii rasprostraneniya tepla pri teploprovodnosti, zavisyaschei ot temperatury”, V sb., posvyaschennom semidesyatiletiyu akademika A. F. Ioffe, AN SSSR, M., 1950

[5] Barenblatt G. I., “O nekotorykh neustanovivshikhsya dvizheniyakh zhidkosti i gaza v poristoi srede”, PMM, 16:1 (1952), 67–78 | MR | Zbl

[6] Barenblatt G. I., “Ob avtomodelnykh resheniyakh zadachi Koshi dlya nelineinogo parabolicheskogo uravneniya nestatsionarnoi filtratsii gaza v poristoi srede”, PMM, 16:6 (1952), 761–763

[7] Barenblatt G. I., “Ob odnom klasse tochnykh reshenii ploskoi odnomernoi zadachi nestatsionarnoi filtratsii gaza v poristoi srede”, PMM, 17:6 (1953), 739–742 | MR | Zbl

[8] Barenblatt G. I., “O predelnykh avtomodelnykh dvizheniyakh v teorii nestatsionarnoi filtratsii gaza v poristoi srede i teorii pogranichnogo sloya”, PMM, 18:4 (1954), 409–414 | MR | Zbl

[9] Ovsyannikov L. V., “Gruppovye svoistva uravneniya nelineinoi teploprovodnosti”, DAN SSSR, 125:3 (1959), 492–495 | Zbl

[10] Samarskii A. A., Sobol I. M., “Primery chislennogo rascheta temperaturnykh voln”, ZhVM i MF, 3:4 (1963), 703–719 | MR

[11] Marshak R. E., “Effect of radiation on shock wave behaviour”, Phys. Fluids, 1 (1958), 24–29 | DOI | MR | Zbl

[12] Pattle R. E., “Diffusion from an instantaneous point source with concentration dependent coefficient”, Quart. J. Mech. Appl. Math., 12 (1959), 407–409 | DOI | MR | Zbl

[13] Serrin J., “Asymptotic behaviour of velocity profilts in the Prandtl boundary layer theory”, Proc. Roy. Soc., A209 (1967), 491–507 | MR

[14] Atkinson F. V., Peletitr L. A., “Similarity profiles of flows through porous media”, Arch. Rat. Mech. Anal., 42 (1971), 369–379 | DOI | MR | Zbl

[15] Peletier L. A., “Asymptotic behaviour of solutions of the porous media equation”, SIAM J. Appl. Math., 21 (1971), 542–551 | DOI | MR | Zbl

[16] Craven A. H., Peletier L. A., “Similarity solutions for degenerate quasilinear parabolic equations”, J. Math. Anal. Appl., 38 (1972), 73–81 | DOI | MR | Zbl

[17] Atkinson F. V., Peletier L. A., “Similarity solutions of the nonlinear diffusion equation”, Arch. Rat. Mech. Anal., 54 (1974), 373–392 | DOI | MR | Zbl

[18] Samarskii A. A., Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P., “Effekt metastabilnoi lokalizatsii tepla v srede s nelineinoi teploprovodnostyu”, DAN SSSR, 223:6 (1975), 1344–1347

[19] Gilding B. H., Peletier L. A., “On a class of similarity solutions of the porous media equation”, J. Math. Anal. Appl., 55 (1976), 351–364 | DOI | MR | Zbl

[20] Gilding B. H., Peletier L. A., “On a class of similarity solutions of the porous media equation. II”, J. Math. Anal. Appl., 57 (1977), 522–538 | DOI | MR | Zbl

[21] Galaktionov V. A., Mikhailov A. P., Ob odnoi avtomodelnoi zadache dlya uravneniya nelineinoi teploprovodnosti, Preprint IPM AN SSSR, No 53, IPM, M., 1977 | MR

[22] Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., Metastabilnaya lokalizatsiya tepla v srede s nelineinoi teploprovodnostyu i usloviya proyavleniya ee v eksperimente, Preprint IPM AN SSSR, No 103, IPM, M., 1977 | MR

[23] Pimbley G. H., “Wave solutions travelling along quadratic path for the equation $(\partial u/\partial t)-(k(u)ux)_x=0$”, Quart. Appl. Math., 35 (1977), 129–138 | MR | Zbl

[24] Duyn C. J., van, Peletier L. A., “Asymptotic behaviour of solutions of a nonlinear diffusion equation”, Arch. Rat. Mech. Anal., 65 (1977), 363–377 | DOI | MR | Zbl

[25] Gilding B. H., “Stabilization of flows through porous media”, SIAM J. Math. Anal., 10 (1979), 237–246 | DOI | MR | Zbl

[26] Babu D. K., Genuchten M. Th., van., “A similarity solution to a nonlinear diffusion equation of the singular type; a uniform valid solution by perturbation”, Quart. Appl. Math., 37 (1979), 11–21 | MR | Zbl

[27] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “Asimptoticheskaya stadiya rezhimov s obostreniem i effektivnaya lokalizatsiya tepla v zadachakh nelineinoi teploprovodnosti”, Diff. uravneniya, 16:7 (1980), 1196–1204 | MR | Zbl

[28] Barenblatt G. I., Vishik M. I., “O konechnoi skorosti rasprostraneniya v zadachakh nestatsionarnoi filtratsii zhidkosti i gaza”, PMM, 20:3 (1956), 411–417 | MR | Zbl

[29] Oleinik O. A., Kalashnikov A. S., Chzhou Yuilin., “Zadacha Koshi i kraevye zadachi dlya uravneniya tipa nestatsionarnoi filtratsii”, Izv. AN SSSR, Ser. matem., 22:5 (1958), 667–704 | MR | Zbl

[30] Sabinina E. S., “Ob odnom klasse nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, DAN SSSR, 143:4 (1962), 794–797 | MR | Zbl

[31] Kalashnikov A. S., “O vozniknovenii osobennostei u reshenii uravnenii nestatsionarnoi filtratsii”, ZhVM i MF, 7:2 (1967), 440–443 | MR

[32] Aronson D. G., “Regularity properties of flows through porous media”, SIAM J. Appl. Math., 17 (1969), 461–467 | DOI | MR | Zbl

[33] Aronson D. G., “Regularity properties of flows through porous media: a counterexample”, SIAM J. Appl. Math., 19 (1970), 299–307 | DOI | MR | Zbl

[34] Aronson D. G., “Regularity properties of flows through porous media: the interface”, Arch. Rat. Mech. Anal., 37 (1970), 1–10 | DOI | MR | Zbl

[35] Kalashnikov A. S., “Ob uravneniyakh tipa nestatsionarnoi filtratsii s beskonechnoi skorostyu rasprostraneniya vozmuschenii”, Vestnik MGU, Ser. matem., mekh., 1972, no. 6, 45–49 | Zbl

[36] Kalashnikov A. S., “O vliyanii pogloscheniya na rasprostranenie tepla v srede s teploprovodnostyu, zavisyaschei ot temperatury”, ZhVM i MF, 1976, no. 3, 689–697

[37] Knerr B. F., “The porous medium eqaution in one dimension”, Trans. Amer. Math. Soc., 234 (1977), 381–415 | DOI | MR

[38] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “Lokalizatsiya tepla v nelineinykh sredakh”, Diff. uravneniya, 17:10 (1981), 1826–1841 | MR | Zbl

[39] Galaktionov V. A., “O priblizhennykh avtomodelnykh resheniyakh uravnenii tipa nelineinoi teploprovodnosti”, Diff. uravneniya, 16:9 (1980), 1660–1676 | MR | Zbl

[40] Galaktionov V. A., Priblizhennye avtomodelnye resheniya uravnenii tipa nelineinoi teploprovodnosti, Preprint IPM AN SSSR, No 106, IPM, M., 1980 | MR

[41] Kamin S., “Similar solutions and the asymptotics of filtration equations”, Arch. Rat. Mech. Anal., 60 (1976), 171–183 | MR | Zbl

[42] Friedman A., Kamin S., “The asymptotic behaviour of gas in an $n$-dimensional porous medium”, Trans. Amer. Math. Soc., 262 (1980), 551–263 | DOI | MR

[43] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “Ob odnom podkhode k sravneniyu reshenii parabolicheskikh uravnenii”, ZhVM i MF, 19:6 (1979), 1451–1461 | MR | Zbl

[44] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., Sravnenie reshenii parabolicheskikh uravnenii na osnove apriornykh potochechnykh otsenok starshei proizvodnoi, Preprint IPM AN SSSR, No 21, IPM, M., 1979 | MR

[45] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “O sravnenii reshenii parabolicheskikh uravnenii”, DAN SSSR, 248:3 (1979), 586–589 | MR | Zbl

[46] Galaktionov V. A., Usloviya $\psi$-kritichnosti i metody sravneniya reshenii parabolicheskikh uravnenii, Preprint IPM AN SSSR, No 151, IPM, M., 1979 | MR

[47] Galaktionov V. A., “Dva metoda sravneniya reshenii parabolicheskikh uravnenii”, Dokl. AN SSSR, 251:4 (1980), 832–835 | MR | Zbl

[48] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Metastabilnaya lokalizatsiya vozmuschenii v zadachakh dlya uravnenii tipa nelineinoi teploprovodnosti, Preprint IPM AN SSSR, No 181, IPM, M., 1979 | MR

[49] Galaktionov V. A., “Priblizhennye avtomodelnye resheniya uravneniya teploprovodnosti”, DAN SSSR, 1982

[50] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., Deistvie granichnykh rezhimov s obostreniem na sredu s postoyannoi teploprovodnostyu, Preprint IPM AN SSSR, No 28, IPM, M., 1979 | MR

[51] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., “Lokalizatsiya protsessov diffvzii v sredakh s postoyannymi svoistvami”, DAN SSSR, 247:2 (1979), 349–353 | MR

[52] Samarskii A. A., “O novykh metodakh issledovaniya asimptoticheskikh svoistv parabolicheskikh uravnenii”, Tr. MIAN, CLVIII, 1981, 153–162 | MR

[53] Sattinger D. H., “Monotone methods in nonlinear elliptic and parabolic boundary value problems”, Indiana Univ. Math. J., 21 (1972), 979–1000 | DOI | MR | Zbl

[54] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[55] Elenin G. G., Kurdyumov S. P., Uslovie uslozhneniya organizatsii nelineinoi dissipativnoi sredy, Preprint IPM AN SSSR, No 106, IPM, M., 1977

[56] Samarskii A. A., Elenin G. G., Zmitrenko N. V., Kurdyumov S. P., Mikhailov A. P., “Gorenie nelineinoi sredy v vide slozhnykh struktur”, DAN SSSR, 237:6 (1977), 1330–1333 | MR

[57] Kurdyumov S. P., Malinetskii G. G., Poveschenko Yu. A., Popov Yu. P., Samarskii A. A., “Vzaimodeistvie dissipativnykh teplovykh struktur v nelineinykh sredakh”, DAN SSSR, 251:4 (1980), 836–839 | MR

[58] Bluman G., Kumei S., “On the remarkable nonlinear diffusion equation $(\partial/\partial x)[a(u+b)^{-2} (\partial u/\partial x)]-(\partial u/\partial t)=0$”, J. Math. Phys., 21 (1980), 1019–1023 | DOI | MR | Zbl

[59] Munier A., Burgan J. R., Gutierrez J., Fijalkow E., Feix M. R., “Group transformations and the nonlinear heat diffusion equation”, SIAM J. Appl. Math., 40 (1981), 191–207 | DOI | MR | Zbl

[60] Caffarelli L. A., Friedman A., “Regularity of the free boundary for the one-dimensional flow of gas in a porous medium”, Amer. J. Math., 101 (1979), 1193–1218 | DOI | MR | Zbl

[61] Caffarelli L. A., Friedman A., “Regularity of the free boundary of a gas flow in an $n$-dimensional porous medium”, Indiana Univ. Math. J., 29 (1980), 361–391 | DOI | MR | Zbl