Geodesic
Initial evolution of supports of solutions of quasilinear parabolic equations with degenerate absorption potential
Sbornik. Mathematics, Tome 204 (2013) no. 3, pp. 383-410 Cet article a éte moissonné depuis la source Math-Net.Ru

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The propagation of supports of solutions of second-order quasilinear parabolic equations is studied; the equations are of the type of nonstationary diffusion, having semilinear absorption with an absorption potential which degenerates on the initial plane. We find sufficient conditions, which are sharp in a certain sense, on the relationship between the boundary regime and the type of degeneration of the potential to ensure the strong localization of solutions. We also establish a weak localization of solutions for an arbitrary potential which degenerates only on the initial plane. Bibliography: 12 titles.
Keywords: quasilinear parabolic equations, absorption potential, strong localization of solutions, weak localization of solutions, energy method. 
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E. V. Stepanova; A. E. Shishkov. Initial evolution of supports of solutions of quasilinear parabolic equations with degenerate absorption potential. Sbornik. Mathematics, Tome 204 (2013) no. 3, pp. 383-410. http://geodesic.mathdoc.fr/item/SM_2013_204_3_a3/

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

[2] S. Antontsev, J. I. Diaz, S. I. Shmarev, “The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms”, Ann. Fac. Sci. Toulouse Math. (6), 4:1 (1995), 5–30 | DOI | MR | Zbl

[3] J. I. Diaz, L. Veron, “Laurent Local vanishing properties of solutions of elliptic and parabolic quasilinear equations”, Trans. Amer. Math. Soc., 290:2 (1985), 787–814 | DOI | MR | Zbl

[4] 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

[5] A. S. Kalashnikov, “On the initial jump of the free boundary in a boundary-value problem for the semilinear heat conduction equation with absorption”, Russian Math. Surveys, 52:6 (1997), 1300–1301 | DOI | DOI | MR | Zbl

[6] S. N. Antontsev, “On the localization of solutions of nonlinear degenerate elliptic and parabolic equations”, Soviet Math. Dokl., 24:2 (1981), 420–424 | MR | Zbl

[7] O. A. Oleinik, G. A. Iosifyan, “Analog printsipa Sen-Venana dlya ellipticheskogo uravneniya vtorogo poryadka i edinstvennost reshenii kraevykh zadach v neogranichennykh oblastyakh”, UMN, 31:4 (1976), 261–262 | MR | Zbl

[8] A. Shishkov, L. Veron, “The balance between diffusion and absorption in semilinear parabolic equations”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18:1 (2007), 59–96 | MR | Zbl

[9] Y. Belaud, A. Shishkov, “Long-time extinction of solutions of some semilinear parabolic equations”, J. Differential Equations, 2007, no. 238, 64–86 | DOI | MR | Zbl

[10] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, de Gruyter, Paris, 1969 | MR | MR | Zbl | Zbl

[11] R. Kersner, A. Shishkov, “Instantaneous shrinking of the support of energy solutions”, J. Math. Anal. Appl., 198:3 (1996), 729–750 | DOI | MR | Zbl

[12] A. E. Shishkov, “Dead cores and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order”, Sb. Math., 190:12 (1999), 1843–1869 | DOI | DOI | MR | Zbl