Blow-up and global existence profile of a class of fully nonlinear degenerate parabolic equations

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1. [1] Allen L.J.S., Persistence and extinction in single-species reaction-diffusion models, Bull. Math. Biol., 1983, 45(2), 209–277 Zbl0543.92020
2. [2] Angenent S., On the formation of singularities in the curve shortening flow, J. Differential Geom., 1991, 33(3), 601–633 Zbl0731.53002
3. [3] Deng K., Levine H.A., The role of critical exponents in blow-up theorems the sequel, J. Math. Anal. Appl., 2000, 243(1), 85–126 http://dx.doi.org/10.1006/jmaa.1999.6663 
4. [4] Duvaut G., Lions J.-L., Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, 21, Dunod, Paris, 1972 Zbl0298.73001
5. [5] Epstein C.L., Weinstein M.I., A stable manifold theorem for the curve shortening equation, Comm. Pure Appl. Math., 1987, 40(1), 119–139 http://dx.doi.org/10.1002/cpa.3160400106 Zbl0602.34026
6. [6] Fujita H., On the blowing up of solutions of the Cauchy problem for u t = Δu+u1+u 1+α , J. Fac. Sci. Univ. Tokyo Sect. I, 1966, 13, 109–124 
7. [7] Galaktionov V.A., Blow-up for quasilinear heat equations with critical Fujita’s exponents, Proc. Roy. Soc. Edinburgh Sect. A, 1994, 124(3), 517–525 Zbl0808.35053
8. [8] Galaktionov V.A., Kurdjumov S.P., Mihaĭlov A.P., Samarskiĭ A.A., On unbounded solutions of the Cauchy problem for the parabolic equation u t = ∇(u σ∇u) + u β, Dokl. Akad. Nauk SSSR, 1980, 252(6), 1362–1364 (in Russian) 
9. [9] Galaktionov V.A., Levine H.A., A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 1998, 34(7), 1005–1027 http://dx.doi.org/10.1016/S0362-546X(97)00716-5 Zbl1139.35317
10. [10] Galaktionov V.A., Pohozaev S.I., Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators, J. Evol. Equ., 2006, 6(1), 45–69 http://dx.doi.org/10.1007/s00028-005-0213-z Zbl1109.35015
11. [11] Levine H.A., The role of critical exponents in blowup theorems, SIAM Rev., 1990, 32(2), 262–288 http://dx.doi.org/10.1137/1032046 Zbl0706.35008
12. [12] Li J., Yin J., Jin C., On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations, Z. Angew. Math. Phys., 2010, 61(5), 835–847 http://dx.doi.org/10.1007/s00033-010-0059-2 Zbl1242.35152
13. [13] Lions P.-L., Some problems related to the Bellman-Dirichlet equation for two operators, Comm. Partial Differential Equations, 1980, 5(7), 753–771 http://dx.doi.org/10.1080/03605308008820153 Zbl0435.35035
14. [14] Low B.C., Resistive diffusion of force-free magnetic fields in a passive medium, Astrophys. J., 1973, 181, 209–226 http://dx.doi.org/10.1086/152042 
15. [15] Low B.C., Resistive diffusion of force-free magnetic fields in a passive medium. II. A nonlinear analysis of the one-dimensional case, Astrophys. J., 1973, 184, 917–929 http://dx.doi.org/10.1086/152382 
16. [16] Mitidieri E., Pohozaev S.I., A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 2001, 3(234), 1–362 Zbl1074.35500
17. [17] Ughi M., A degenerate parabolic equation modelling the spread of an epidemic, Ann. Mat. Pura Appl., 1986, 143, 385–400 http://dx.doi.org/10.1007/BF01769226 Zbl0617.35066
18. [18] Wang L., On the regularity theory of fully nonlinear parabolic equations, Bull. Amer. Math. Soc. (N.S.), 1990, 22(1), 107–114 http://dx.doi.org/10.1090/S0273-0979-1990-15854-9 Zbl0704.35025
19. [19] Winkler M., A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 2002, 25(11), 911–925 http://dx.doi.org/10.1002/mma.319 Zbl1007.35043
20. [20] Yin J., Li J., Jin C., Classical solutions for a class of fully nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 2009, 360(1), 119–129 http://dx.doi.org/10.1016/j.jmaa.2009.06.038 Zbl1173.35567