﻿ 多重非线性抛物方程组解的爆破 Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations

Pure Mathematics
Vol.05 No.02(2015), Article ID:14934,6 pages
10.12677/PM.2015.52009

Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations

Jing Su, Longfei Qi, Qingying Hu

College of Science, Henan University of Technology, Zhengzhou Henan

Email: slxhqy@163.com

Received: Feb. 27th, 2015; accepted: Mar. 8th, 2015; published: Mar. 12th, 2015

Copyright © 2015 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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.

Keywords:Blowup of Solution, Doubly Nonlinear Parabolic Equations, Levine’s Concavity Method

Email: slxhqy@163.com

1. 引言

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

Sacks [18] 研究了如下包含方程解的爆破问题

(1.7)

Zhang [19] 和Ding和Guo [20] -[22] 通过构造适当的辅助函数，利用一阶微分不等式考虑了下面带梯度项和Neumann (或Robin)初边值问题解的爆破条件

(1.8)

Korpusou和Sveshnikov [23] [24] 给出了如下方程初边值问题弱解爆破的充分条件

(1.9)

2. 假设和基本引理

(A1)，存在函数使得

,

，即

,

,

, ,.

3. 主要结果及证明

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

,

(3.11)

(3.12)

(3.13)

(3.13)关于t积分得

(3.14)

(3.15)

(3.12)结合(3.15)，并用到，得

(3.16)

(3.17)

(3.18)

, ,

Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations. 理论数学,02,59-65. doi: 10.12677/PM.2015.52009

1. 1. Kalashnikov, A.S. (1987) Some problem of the qualitative theory of nonlinear degenerate second-order parabolic equa-tions. Russian Mathematical Surveys, 42, 169-222.

2. 2. Lions, J.L. (1969) Quelques methodes de resolution desprob-lemes aux limites non lineaires. Dunod, Paris.

3. 3. Ivanov, A.V. (1993) Quasilinear parabolic equations admitting double degeneracy. St. Petersburg Mathematical Journal, 4, 1153-1168.

4. 4. Laptev, G.I. (1997) Solvability of second-order quasilinear parabolic equations with double degeneration. Siberian Mathematical Journal, 38, 1160-1177.

5. 5. Tsutsumi, M. (1988) On solution of some doubly nonlinear parabolic equations with absorption. Journal of Mathematical Analysis and Applications, 132, 187-212.

6. 6. Laptev, G.I. (2000) Evolution equations with monotone operator and functional nonlinearity at the time derivative. Sbornik: Mathematics, 19, 1301-1322.

7. 7. Eden, A. and Rakotoson, J.M. (1994) Exponential attractors for a doubly nonlinear equation. Journal of Mathematical Analysis and Applications, 185, 321-339

8. 8. 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

9. 9. Miranville, A. (2006) Finite dimensional global attractor for a class of doubly nonlinear parabolic equation. Central European Journal of Mathematics, 4, 163-182.

10. 10. Miranville, A. and Zelik, S. (2007) Finite-dimensionality of attractors for degenerate equations of elliptic–parabolic type. Nonlinearity, 20, 1773-1797.

11. 11. 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.

12. 12. 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.

13. 13. 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.

14. 14. 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.

15. 15. 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.

16. 16. 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.

17. 17. 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.

18. 18. Sacks, P.E. (1983) Continuity of solutions of a singular parabolic equation. Nonlinear Analysis, TMA, 7, 387-409.

19. 19. Zhang, H.L. (2008) Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Analysis, 69, 4567-4575.

20. 20. 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.

21. 21. 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.

22. 22. 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.

23. 23. Korpusov, M.O. and Sveshnikov, A.G. (2011) Blowup in nonlinear Sobolev type equation. De Gruyter, Berlin/New York.

24. 24. Korpusov, M.O. (2013) Solution blow-up for a class of parabolic equations with double nonlinearity. Sbornik: Mathematics, 204, 323-346.

25. 25. 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.

26. 26. Ouardi, H.E. (2007) On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories. Archivum Mathematicum, 43, 289-303.

27. 27. Agre, K. and Rammaha, M.A. (2006) Systems of nonlinear wave equations with damping and source terms. Differential and Integral Equations, 19, 1235-1270.

28. 28. 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.