Geodesic
Fujita type theorems for quasilinear parabolic equations
Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 459-478 Cet article a éte moissonné depuis la source Math-Net.Ru

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This work deals with the Cauchy problem for a parabolic equation with a double non-linearity of the following type: $$ u_t=\operatorname{div}(u^\alpha|Du|^{m-1}Du)+u^p, $$ where $0. Existence and non-existence results for global solutions of this problem with initial conditions that slowly decay to zero are established. 
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     author = {N. V. Afanasieva and A. F. Tedeev},
     title = {Fujita type theorems for quasilinear parabolic equations},
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     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_4_a0/}
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N. V. Afanasieva; A. F. Tedeev. Fujita type theorems for quasilinear parabolic equations. Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 459-478. http://geodesic.mathdoc.fr/item/SM_2004_195_4_a0/

[1] Kalashnikov A. S., “Nekotorye voprosy kachestvennoi teorii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, UMN, 42:2(254) (1987), 135–176 | MR | Zbl

[2] Di Benedetto E., Herrero M. A., “Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1

2$”, Arch. Ration. Mech. Anal., 111 (1990), 225–290 | DOI | MR

[3] Vespri V., “On the local behavior of solutions of a certain class of doubly nonlinear parabolic equations”, Manuscripta Math., 75 (1992), 65–80 | DOI | MR | Zbl

[4] Tsutsumi M., “On solution of some doubly nonlinear parabolic equations with absorbtion”, J. Math. Anal. Appl., 132 (1988), 187–212 | DOI | MR | Zbl

[5] Kaplan S., “On the grouth of solutions of quasilinear parabolic equations”, Comm. Pure Appl. Math., 16 (1963), 305–330 | DOI | MR | Zbl

[6] Fujita H., “On the blowing up of solutions to the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$”, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124 | MR | Zbl

[7] Hayakawa K., “On nonexistence of global solutions of some semilinear parabolic differential equations”, Proc. Japan Acad. Ser. A Math. Sci., 49 (1973), 503–505 | MR | Zbl

[8] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[9] Galaktionov V. A., Levine H. A., “A general approach to critical Fujita exponents and systems”, Nonlinear Anal., 34 (1998), 1005–1027 | DOI | MR | Zbl

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

[11] Andreucci D., Tedeev A. F., “Optimal bounds and blow-up phenomena for parabolic problems in narrowing domains”, Proc. Roy. Soc. Edinburg Sect. A, 128:6 (1998), 1163–1180 | MR | Zbl

[12] Liu X., Wang M., “The critical exponent of doubly singular parabolic equations”, J. Math. Anal. Appl., 257:1 (2001), 170–188 | DOI | MR | Zbl

[13] Mukai K., Mochuzuki K., Huang Q., “Large time behavior and life span for a quasilinear parabolic equation with slow decay initial values”, Nonlinear Anal., 39A:1 (2000), 33–45 | DOI | MR

[14] Kamin S., Ughi M., “On the behavior as $t\to\infty$ of the solutions the Cauchy problem for certain nonlinear parabolic equations”, J. Math. Anal. Appl., 128 (1987), 456–469 | DOI | MR | Zbl

[15] Deng K., Levine H. A., “The role of critical exponents in blow-up theorems: The sequel”, J. Math. Anal. Appl., 243 (2000), 85–126 | DOI | MR | Zbl

[16] Levine H. A., “The role of critical exponents in blow up theorems”, SIAM Rev., 32 (1990), 262–288 | DOI | MR | Zbl

[17] Cirmi G. R., Leonardi S., Tedeev A. F., The asymptotic behavior of the solution of a quasilinear parabolic equation with blow-up term, Preprint, Univ. Catania Dipart. Matem. Inform., Catania, 1998

[18] Andreucci D., Di Benedetto E., “On the Cauchy problem and initial traces for a class of evolution equaions with strongly nonlinear sources”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 363–441 | MR | Zbl

[19] Andreucci D., “Degenerate parabolic equations with initial data measares”, Trans. Amer. Math. Soc., 340:10 (1997), 3911–3923 | DOI | MR

[20] Junning Z., “On the Cauchy problem and initial traces for the evolution $p$t-Laplacian equations with strongly nonlinear sources”, J. Differential Equations, 121 (1995), 329–383 | DOI | MR | Zbl

[21] Ishige K., “On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation”, SIAM J. Math. Anal., 27:5 (1996), 1235–1260 | DOI | MR | Zbl

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