Geodesic
Local and Global Estimates of the Solutions of the Cauchy Problem for Quasilinear Parabolic Equations with a Nonlinear Operator of Baouendi--Grushin Type
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 395-407.

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In this paper, the qualitative properties of the solutions of the Cauchy problem for degenerate parabolic equations with a nonlinear operator of Baouendi–Grushin type are studied. Sharp local and global (with respect to the spatial and temporal variables) estimates of the solution are obtained. The property of the finiteness of the support of the solution is established.
Keywords: quasilinear parabolic equation, nonlinear operator, Cauchy problem, Carnot–Carathéodory space, Young's inequality, maximum principle. 
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V. A. Markasheva; A. F. Tedeev. Local and Global Estimates of the Solutions of the Cauchy Problem for Quasilinear Parabolic Equations with a Nonlinear Operator of Baouendi--Grushin Type. Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 395-407. http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a7/

[1] A. S. Kalashnikov, “Nekotorye voprosy kachestvennoi teorii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, UMN, 42:2 (1987), 135–176 | MR | Zbl

[2] V. V. Grushin, “Ob odnom klasse gipoellipticheskikh operatorov”, Matem. sb., 83:3 (1970), 456–473 | MR | Zbl

[3] M. S. Baouendi, “Sur une classe d'opérateurs elliptiques dégénérés”, Bull. Soc. Math. France, 95 (1967), 45–87 | MR | Zbl

[4] B. Franchi, E. Lanconelli, “Une métrique associée à une classe d'opérateurs elliptiques dégénérés”, Conference on Linear Partial and Pseudodifferential Operators (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino, 1983, Special Issue, Univ. Politec. Torino, Torino, 1984, 105–114 | MR | Zbl

[5] B. Franchi, E. Lanconelli, “Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10:4 (1983), 523–541 | MR | Zbl

[6] N. Garofalo, D.-M. Nhieu, “Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces”, Comm. Pure Appl. Math., 49:10 (1996), 1081–1144 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[7] E. Di Benedetto, M. A. Herrero, “On the Cauchy problem and initial traces for a degenerate parabolic equation”, Trans. Amer. Math. Soc., 314:1 (1989), 187–224 | DOI | MR | Zbl

[8] S. N. Antontsev, “O lokalizatsii reshenii nelineinykh vyrozhdayuschikhsya ellipticheskikh i parabolicheskikh uravnenii”, Dokl. AN SSSR, 260:6 (1981), 1289–1293 | MR | Zbl

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

[10] D. Andreucci, A. F. Tedeev, “Universal bounds at the blow-up time for nonlinear parabolic equations”, Adv. Differential Equations, 10:1 (2005), 89–120 | MR | Zbl

[11] D. Andreucci, A. F. Tedeev, “Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity”, Adv. Differential Equations, 5:7–9 (2000), 833–860 | MR | Zbl

[12] D. Andreucci, A. F. Tedeev, “Finite speed of propagation for thin-film equations and other higher-order parabolic equations with general nonlinearity”, Interfaces Free Bound., 3:3 (2001), 233–264 | MR | Zbl

[13] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Fizmatlit, M., 1967 | MR | Zbl

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