The Cauchy problem for a quasilinear parabolic equation with gradient absorption

V. A. Markasheva   ; An. F. Tedeev

Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 581-611

Résumé

The qualitative properties of solutions to the Cauchy problem for a degenerate parabolic equation containing a nonlinear operator of Baouendi-Grushin type and with gradient absorption whose density depends on time, as well as the space variables, are investigated. Bounds for the diameter of the support of the solution which are sharp with respect to time are obtained, together with its maximum. A condition which determines whether or not the phenomenon of decay to zero of the total mass of the solution occurs is discovered. Bibliography: 35 titles.

Détail

Keywords: operator of Baouendi-Grushin type, quasilinear parabolic equation, decay of the total mass of a solution, estimate for the support of the solution.
Mots-clés : gradient absorption

Bibliographie

[1] N. Garofalo, D.-M. Nhieu, “Isoperimetric and Sobolev inequalities for Carnot–Caratheodory 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

[2] I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi–Grushin vector fields, 2007, arXiv: 0704.1343

[3] 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

[4] V. V. Grušin, “On a class of hypoelliptic operators”, Math. USSR-Sb., 12:3 (1970), 458–476 | DOI | MR | Zbl | Zbl

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

[6] B. Franchi, E. Lanconelli, “Une métrique associée à une classe d'opérateurs elliptiques degeneres”, Conference on linear partial and pseudodifferential operators (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino, Università e Politecnico, Toronto, 1984, 105–114 | MR | Zbl

[7] 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

[8] 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”, Math. Notes, 85:3 (2009), 385–396 | DOI | MR | Zbl

[9] V. A. Markasheva, “Lokalnaya gëlderovost reshenii kvazilineinykh parabolicheskikh uravnenii s nelineinym operatorom tipa Baouendi–Grushina, I”, Tr. IPMM, 16, IPMM NAN Ukrainy, Donetsk, 2008, 124–135 | MR

[10] V. A. Markasheva, “Lokalnaya gëlderovost reshenii kvazilineinykh parabolicheskikh uravnenii s nelineinym operatorom tipa Baouendi–Grushina, II”, Tr. IPMM, 17, IPMM NAN Ukrainy, Donetsk, 2008, 128–143 | MR

[11] S. Benachour, B. Roynette, P. Vallois, “Asymptotic estimates of solutions of $u_t-\frac{1}{2}\Delta u=-|\nabla u|$ in $R_+\times R^d$, $d\ge 2$”, J. Funct. Anal., 144:2 (1997), 301–324 | DOI | MR | Zbl

[12] Th. Gallay, Ph. Laurençot, “Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent”, Indiana Univ. Math. J., 56:1 (2007), 459–479 | DOI | MR | Zbl

[13] M. Ben-Artzi, “Global existence and decay for a nonlinear parabolic equation”, Nonlinear Anal., 19:8 (1992), 763–768 | DOI | MR | Zbl

[14] G. Barles, F. Da Lio, “On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations”, J. Math. Pures Appl. (9), 83:1 (2004), 53–75 | DOI | MR | Zbl

[15] L. Amour, M. Ben-Artzi, “Global existence and decay for viscous Hamilton–Jacobi equations”, Nonlinear Anal., 31:5–6 (1998), 621–628 | DOI | MR | Zbl

[16] S. Benachour, Ph. Laurençot, “Global solutions to viscous Hamilton-Jacobi equations with irregular initial data”, Comm. Partial Differential Equations, 24:11–12 (1999), 1999–2021 | DOI | MR | Zbl

[17] M. Ben-Artzi, H. Koch, “Decay of mass for a semilinear parabolic equation”, Comm. Partial Differential Equations, 24:5–6 (1999), 869–881 | DOI | MR | Zbl

[18] A. Dall'Aglio, D. Giachetti, S. Segura de León, “Global existence for parabolic problems involving the $p$-Laplacian and a critical gradient term”, Indiana Univ. Math. J., 58:1 (2009), 1–48 | DOI | MR | Zbl

[19] Ph. Laurençot, J. Vázquez, “Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption”, J. Dynam. Differential Equations, 19:4 (2007), 985–1005 | DOI | MR | Zbl

[20] S. Snoussi, S. Tayachi, “Large time behavior of solutions for parabolic equations with nonlinear gradient terms”, Hokkaido Math. J., 36:2 (2007), 311–344 | MR | Zbl

[21] J.-Ph. Bartier, Ph. Laurençot, “Gradient estimates for a degenerate parabolic equation with gradient absorption and applications”, J. Funct. Anal., 254:3 (2008), 851–878 | DOI | MR | Zbl

[22] Ch. Stinner, “Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion”, J. Differential Equations, 248:2 (2010), 209–228 | DOI | MR | Zbl

[23] D. Andreucci, A. F. Tedeev, M. Ughi, “The Cauchy problem for degenerate parabolic equations with source and damping”, Ukr. Math. Bull., 1:1 (2004), 1–23 | MR | Zbl

[24] A. F. Tedeev, “Initial-boundary value problems for quasilinear degenerate parabolic equations with damping. The Neumann problem”, Ukrainian Math. J., 58:2 (2006), 304–317 | DOI | MR | Zbl

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

[26] 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

[27] D. Andreucci, A. F. Tedeev, “A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary”, J. Math. Anal. Appl., 231:2 (1999), 543–567 | DOI | MR | Zbl

[28] D. S. Jerison, “The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II”, J. Funct. Anal., 43:2 (1981), 224–257 | DOI | MR | Zbl

[29] S. K. Vodopyanov, “Foundations of the theory of mappings with bounded distortion on Carnot groups”, The interaction of analysis and geometry, Contemp. Math., 424, Amer. Math. Soc, Providence, RI, 2007, 303–344 | DOI | MR | Zbl

[30] E. A. Plotnikova, “Integral representations and the generalized Poincare inequality on Carnot groups”, Siberian Math. J., 49:2 (2008), 339–352 | DOI | MR | Zbl

[31] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[32] E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993 | MR | Zbl

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

[34] D. Andreucci, “Degenerate parabolic equations with initial data measures”, Trans. Amer. Math. Soc., 349:10 (1997), 3911–3923 | DOI | MR | Zbl

[35] H. Shang, F. Li, “On the Cauchy problem for the evolution $p$-Laplacian equations with gradient term and source and measures as initial data”, Nonlinear Anal., 72:7–8 (2010), 3396–3411 | DOI | MR | Zbl