Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations
Abstract: This paper is concerned with a class of doubly nonlinear parabolic systems. Under the homogeneous Dirichlet conditions and suitable conditions on the nonlinearity and certain initial datum, a sufficient condition for finite time blowup of its solution in a bounded domain is gave by using a modification of Levine’s concavity method.
文章引用: 苏 璟 , 齐龙飞 , 呼青英 (2015) 多重非线性抛物方程组解的爆破。 理论数学， 5， 59-65. doi: 10.12677/PM.2015.52009
 Kalashnikov, A.S. (1987) Some problem of the qualitative theory of nonlinear degenerate second-order parabolic equa-tions. Russian Mathematical Surveys, 42, 169-222.
 Lions, J.L. (1969) Quelques methodes de resolution desprob-lemes aux limites non lineaires. Dunod, Paris.
 Ivanov, A.V. (1993) Quasilinear parabolic equations admitting double degeneracy. St. Petersburg Mathematical Journal, 4, 1153-1168.
 Laptev, G.I. (1997) Solvability of second-order quasilinear parabolic equations with double degeneration. Siberian Mathematical Journal, 38, 1160-1177.
 Tsutsumi, M. (1988) On solution of some doubly nonlinear parabolic equations with absorption. Journal of Mathematical Analysis and Applications, 132, 187-212.
 Laptev, G.I. (2000) Evolution equations with monotone operator and functional nonlinearity at the time derivative. Sbornik: Mathematics, 19, 1301-1322.
 Eden, A. and Rakotoson, J.M. (1994) Exponential attractors for a doubly nonlinear equation. Journal of Mathematical Analysis and Applications, 185, 321-339
 Eden, A., Michaux, B. and Rakotoson, J.M. (1991) Doubly nonlinear parabolic type equations as dynamical systems. Journal of Dynamics and Differential Equations, 3, 87-131
 Miranville, A. (2006) Finite dimensional global attractor for a class of doubly nonlinear parabolic equation. Central European Journal of Mathematics, 4, 163-182.
 Miranville, A. and Zelik, S. (2007) Finite-dimensionality of attractors for degenerate equations of elliptic–parabolic type. Nonlinearity, 20, 1773-1797.
 Ouardi, H.E. and Hachimi, A.E. (2001) Existence and attractors of solutions for nonlinear parabolic systems. Electronic Journal of Qualitative Theory of Differential Equations, 2001, 1-16.
 Ouardi, H.E. and Hachimi, A.E. (2006) Attractors for a class of doubly nonlinear parabolic systems. Electronic Journal of Qualitative Theory of Differential Equations, 2006, 1-15.
 Levine, H.A. (1973) Some nonexistence and instability theorems for solutions of formally parabolic equations of the form . Archive for Rational Mechanics and Analysis, 51, 371-386.
 Levine, H.A., Park, S.R. and Serrin, J.M. (1998) Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. Journal of Differential Equations, 142, 212-229.
 Iami, T. and Mochizuki, K. (1991) On the blowup of solutions for quasilinear degenerate parabolic equations. Publications of the Research Institute for Mathematical Sciences (Kyoto University), 27, 695-709.
 Levine, H.A. and Sacks, P.E. (1984) Some existence and nonexistence theorems for solutions of degenerate parabolic equations. Journal of Differential Equations, 52, 135-161.
 Levine, H.A. and Payne, L.E. (1974) Nonexistence theorems for the heat equations with nonlinear boundary conditions and for the porous medium equation backward in time. Journal of Differential Equations, 16, 319-334.
 Sacks, P.E. (1983) Continuity of solutions of a singular parabolic equation. Nonlinear Analysis, TMA, 7, 387-409.
 Zhang, H.L. (2008) Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Analysis, 69, 4567-4575.
 Ding, J.T. and Guo, B.Z. (2009) Global and blowup solutions for nonlinear parabolic equations with a gradient term. Houston Journal of Mathematics, 37, 1265-1277.
 Ding, J.T. and Guo, B.Z. (2011) Blow-up solution of nonlinear reaction-diffusion equations under boundary feedback. Journal of Dynamical and Control Systems, 17, 273-290.
 Ding, J.T. (2013) Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions. Computers and Mathematics with Applications, 65, 1808-1822.
 Korpusov, M.O. and Sveshnikov, A.G. (2011) Blowup in nonlinear Sobolev type equation. De Gruyter, Berlin/New York.
 Korpusov, M.O. (2013) Solution blow-up for a class of parabolic equations with double nonlinearity. Sbornik: Mathematics, 204, 323-346.
 Hachimi, A.E. and Ouardi, H.E. (2002) Existence and of regularity a global attractor for doubly nonlinear parabolic equations. Electronic Journal of Differential Equations, 2002, 1-15.
 Ouardi, H.E. (2007) On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories. Archivum Mathematicum, 43, 289-303.
 Agre, K. and Rammaha, M.A. (2006) Systems of nonlinear wave equations with damping and source terms. Differential and Integral Equations, 19, 1235-1270.
 Korpusov, M.O. (2012) Blow-up for a positive-energy solution of model wave equations in nonlinear dynamics. Theoretical and Mathematical Analysis, 171, 421-434.