Estimates of the stabilization rate as $t\to\infty$ of solutions of the first mixed problem for a quasilinear system of second-order parabolic equations

L. M. Kozhevnikova   ; F. Kh. Mukminov

Sbornik. Mathematics, Tome 191 (2000) no. 2, pp. 235-273

Résumé

A quasilinear system of parabolic equations with energy inequality is considered in a cylindrical domain $\{t>0\}\times\Omega$. In a broad class of unbounded domains $\Omega$ two geometric characteristics of a domain are identified which determine the rate of convergence to zero as $t\to\infty$ of the $L_2$-norm of a solution. Under additional assumptions on the coefficients of the quasilinear system estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation.

Bibliographie

[1] Poper F. O., “O stabilizatsii resheniya zadachi Koshi dlya parabolicheskogo uravneniya s peremennymi koeffitsientami”, Dokl. AN SSSR, 153 (1963), 273–275 | MR

[2] Poper F. O., Eidelman S. D., “Teoremy o blizosti reshenii parabolicheskikh uravnenii i stabilizatsiya reshenii zadachi Koshi”, Dokl. AN SSSR, 221 (1975), 32–35 | MR

[3] Zhikov V. V., “O stabilizatsii reshenii parabolicheskikh uravnenii”, Matem. sb., 104(146) (1977), 597–616 | MR | Zbl

[4] Kamin S., “On stabilization of solutions of the Cauchy problem for parabolic equations”, Proc. Roy. Soc. Edinburgh Sect. A, 76:1 (1976), 43–53 | MR | Zbl

[5] Guschin A. K., “O ravnomernoi stabilizatsii reshenii vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 119(161):4(12) (1982), 451–508 | MR

[6] Guschin A. K., Mikhailov V. P., Mikhailov Yu. A., “O ravnomernoi stabilizatsii resheniya vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 128 (170) (1985), 147–168 | MR | Zbl

[7] Mukminov F. Kh., “O ravnomernoi stabilizatsii reshenii pervoi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 181:11 (1990), 1486–1509 | MR

[8] Guschin A. K., “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Tr. MIAN, 126, Nauka, M., 1973, 5–45 | MR | Zbl

[9] Guschin A. K., “Stabilizatsiya reshenii vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 101(143) (1976), 459–499 | MR | Zbl

[10] Lezhnev A. V., “O povedenii pri bolshikh znacheniyakh vremeni neotritsatelnykh reshenii vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 129(171):2 (1986), 186–200 | MR

[11] Tedeev A. F., “Otsenki skorosti stabilizatsii pri $t\to\infty$ resheniya vtoroi smeshannoi zadachi dlya kvazilineinogo parabolicheskogo uravneniya vtorogo poryadka”, Differents. uravneniya, 27:10 (1991), 1795–1806 | MR

[12] Ushakov V. I., “Stabilizatsiya reshenii tretei smeshannoi zadachi dlya parabolicheskogo uravneniya v netsilindricheskoi oblasti”, Matem. cb., 111(153) (1980), 95–115 | MR | Zbl

[13] Bogoyavlenskii O. V., Vladimirov V. S., Volovich I. V., Guschin A. K., Drozhzhinov Yu. N., Zharinov V. V., Mikhailov V. P., “Kraevye zadachi matematicheskoi fiziki”, Tr. MIAN, 175, Nauka, M., 1986, 63–102 | MR | Zbl

[14] Mukminov F. Kh., Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya sistemy uravnenii Nave–Stoksa, Diss. $\dots$ dokt. fiz.-matem. nauk, MIRAN, M., 1994

[15] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[16] Mukminov F. Kh., “Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 111(153):4 (1980), 503–521 | MR | Zbl

[17] Oleinik O. A., Iosifyan G. A., “Analog printsipa Sen-Venana i edinstvennost reshenii kraevykh zadach v neogranichennykh oblastyakh dlya parabolicheskikh uravnenii”, UMN, 31 (1976), 142–166 | MR | Zbl

[18] Tedeev A. F., “Stabilizatsiya reshenii pervoi smeshannoi zadachi dlya kvazilineinogo parabolicheskogo uravneniya vysokogo poryadka”, Differents. uravneniya, 25:3 (1989), 491–498 | MR | Zbl

[19] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[20] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[21] Petrovskii I. G., Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1964 | MR | Zbl

[22] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | MR

[23] Moser J., “A Harnack inequality for parabolic differential equations”, Comm. Pure Appl. Math., 17 (1964), 101–134 | DOI | MR | Zbl

[24] Nash J., “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–953 | DOI | MR