Solutions to higher-order anisotropic parabolic equations in unbounded domains

L. M. Kozhevnikova   ; A. A. Leont'ev

Sbornik. Mathematics, Tome 205 (2014) no. 1, pp. 7-44

Résumé

The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder $D=(0,\infty)\times\Omega$, where $\Omega\subset\mathbb R^n$, $n\geqslant 3$, is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as $t\to \infty$ is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently ‘narrow’. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.

Détail

Keywords: higher-order anisotropic equation, parabolic equation with double nonlinearity, rate of decay of a solution.
Mots-clés : existence of a solution

Bibliographie

[1] È. R. Andriyanova, F. Kh. Mukminov, “Stabilization of the solution of a doubly nonlinear parabolic equation”, Sb. Math., 204:9 (2013), 1239–1263 | DOI | DOI

[2] L. M. Kozhevnikova, F. Kh. Mukminov, “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”, Sb. Math., 191:2 (2000), 235–273 | DOI | DOI | MR | Zbl

[3] L. M. Kozhevnikova, “Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation”, Sb. Math., 196:7 (2005), 999–1032 | DOI | DOI | MR | Zbl

[4] R. Kh. Karimov, L. M. Kozhevnikova, “Stabilization of solutions of quasilinear second order parabolic equations in domains with non-compact boundaries”, Sb. Math., 201:9 (2010), 1249–1271 | DOI | DOI | MR | Zbl

[5] S. P. Degtyarev, A. F. Tedeev, “$L_1$–$L_\infty$ estimates of solutions of the Cauchy problem for an anisotropic degenerate parabolic equation with double non-linearity and growing initial data”, Sb. Math., 198:5 (2007), 639–660 | DOI | DOI | MR | Zbl

[6] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Etudes mathématiques, Dunod–Gauthier-Villars, Paris, 1969, xx+554 pp. | MR | MR | Zbl | Zbl

[7] P. A. Raviart, “Sur la résolution de certaines equations paraboliques non linéaires”, J. Functional Analysis, 5:2 (1970), 299–328 | DOI | MR | Zbl

[8] H. W. Alt, S. Luckhaus, “Quasilinear elliptic-parabolic differential equations”, Math. Z., 183:3 (1983), 311–341 | DOI | MR | Zbl

[9] A. S. Kalashnikov, “Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations”, Russian Math. Surveys, 42:2 (1987), 169–222 | DOI | MR | Zbl

[10] F. Bernis, “Existence results for doubly nonlinear higher order parabolic equations on unbounded domains”, Math. Ann., 279:3 (1988), 373–394 | DOI | MR | Zbl

[11] O. Grange, F. Mignot, “Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires”, J. Functional Analysis, 11:1 (1972), 77–92 | DOI | MR | Zbl

[12] M. Bendahmane, K. H. Karlsen, “Nonlinear anisotropic elliptic and parabolic equations in $\mathbb{R}^N$ with advection and lower order terms and locally integrable data”, Potential Anal., 22:3 (2005), 207–227 | DOI | MR | Zbl

[13] A. Bamberger, “Étude d'une équation doublement non linéaire”, J. Functional Analysis, 24:2 (1977), 148–155 | DOI | MR | Zbl

[14] A. V. Ivanov, P. Z. Mkrtychyan, “Existence of Hölder continuous generalized solutions of the first boundary value problem for quasilinear doubly degenerate parabolic equations”, J. Soviet Math., 62:3 (1992), 2725–2740 | DOI | MR | Zbl

[15] A. V. Ivanov, “Quasilinear parabolic equations that admit double degeneration”, St. Petersburg Math. J., 4:6 (1993), 1153–1168 | MR | Zbl

[16] G. I. Laptev, “Solvability of second-order quasilinear parabolic equations with double degeneration”, Siberian Math. J., 38:6 (1997), 1160–1177 | DOI | MR | Zbl

[17] G. I. Laptev, “Weak solutions of second-order quasilinear parabolic equations with double non-linearity”, Sb. Math., 188:9 (1997), 1343–1370 | DOI | DOI | MR | Zbl

[18] G. I. Laptev, “Evolution equations with monotone operator and functional non-linearity at the time derivative”, Sb. Math., 191:9 (2000), 1301–1322 | DOI | DOI | MR | Zbl

[19] E. R. Andriyanova, F. Kh. Mukminov, “Otsenka snizu skorosti ubyvaniya resheniya parabolicheskogo uravneniya s dvoinoi nelineinostyu”, Ufimsk. matem. zhurn., 3:3 (2011), 3–14 | Zbl

[20] L. M. Kozhevnikova, A. A. Leontev, “Otsenki resheniya anizotropnogo parabolicheskogo uravneniya s dvoinoi nelineinostyu”, Ufimsk. matem. zhurn., 3:4 (2011), 64–85 | Zbl

[21] L. M. Kozhevnikova, A. A. Leontiev, “Decay of solution of anisotropic doubly nonlinear parabolic equation in unbounded domains”, Ufa Math. J., 5:1 (2013), 63–82 | DOI

[22] A. F. Tedeev, “Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations”, Ukrainian Math. J., 44:10 (1992), 1325–1334 | DOI | MR | Zbl

[23] N. D. Alikakos, R. Rostamian, “Gradient estimates for degenerate diffusion equations. II”, Proc. Roy. Soc. Edinburgh Sect. A, 91:3-4 (1981/1982), 335–346 | DOI | MR | Zbl

[24] L. M. Kozhevnikova, F. Kh. Mukminov, “Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains”, Proc. Steklov Inst. Math., 278:1 (2012), 106–120 | DOI | MR

[25] L. M. Kozhevnikova, “Stabilization of solutions of pseudo-differential parabolic equations in unbounded domains”, Izv. Math., 74:2 (2010), 325–345 | DOI | DOI | MR | Zbl

[26] V. Trénoguine, Analyse fonctionnelle, Traduit Russe Math., Mir, Moscow, 1985, 528 pp. | MR | MR | Zbl

[27] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral representations of functions and imbedding theorems, v. I, II, Scripta Series in Mathematics, V. H. Winston Sons, Washington, D.C.; Halsted Press [John Wiley Sons], New York–Toronto, Ont.–London, 1978, 1979, viii+345 pp., viii+311 pp. | MR | MR | MR | Zbl | Zbl

[28] G. Sansone, Equazioni differenziali nel campo reale, v. 2, 2nd ed., N. Zanichelli, Bologna, 1949, xvi+475 pp. | MR | Zbl

[29] L. Nirenberg, “On elliptic partial differentional equations”, Ann. Scuola Norm. Sup. Pisa (3), 13:2 (1959), 115–162 | MR | Zbl