AIMS Mathematics, 2017, 2(4): 647-657. doi: 10.3934/Math.2017.4.647

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Blow-up phenomena for a class of metaparabolic equations with time dependent coeffcient

1 School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, P.R. China
2 Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong, 510006, P.R. China

This paper deals with the initial boundary value problem for a metaparabolic equations withtime dependent coeffcient. Under suitable conditions on initial data, a blow-up criterion which ensuresthat u cannot exist all time is given, and an upper bound for blow up time is derived. Moreover, wealso obtain a lower bound for blow-up time if blow up does occur by means of a di erential inequalitytechnique.
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1. P. M. Brown , Constructive function-theoretic methods for fourth order pseudo-parabolic and metaparabolic equations, Thesis, Indiana University, Bloomington, Indiana, 1973.

2. R. P. Gilbert and G. C. Hsiao, Constructive function theoretic methods for higher order pseudoparabolic equations, Function Theoretic Methods for Partial Di erential Equations, Lect. Notes Math., Springer, Berlin, 561 (1976): 51-67.    

3. E. C. Aifantis, On the problem of diffusion in solids, Acta. Mech., 37 (1980): 265-296.    

4. K. Kuttler and E. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM J. Math. Anal., 19 (1988): 110-120.    

5. T. W. Ting, A cooling process according to two temperature theory of heat conduction, J. Math. Anal. Appl., 46 (1974): 23-31.    

6. Y. D. Shang, Blow-up of solutions for the nonlinear Sobolev-Galpern equations, Mathematica Applicata (Chiness), 13 (2000): 35-39.

7. R. E. Showalter, Sobolev equations for nonlinear dispersive systems, Appl. Anal., 7 (1978): 297-308.    

8. A. T. Bui, Nonlinear evolution equations of Sobolev-Galpern type, Math. Z., 151 (1976): 219-233.    

9. Y. C. Liu and F.Wang, A class of multi-dimensional nonlinear Sobolev-Galpern equations, Acta. Math. Appl. Sinica (Chiness), 17 (1994): 569-577.

10. Y. Y. Ke and J. X. Yin, A note on the viscous Cahn-Hilliard equation, Northeast Math. J., 20 (2004): 101-108.

11. C. M. Elliott and I. N. Kostin, Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn-Hilliard equation, Nonlinearity, 9 (1996): 687-702.    

12. F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation, part I: computations, Nonlinearity, 8 (1995): 131-160.    

13. C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996): 404-423.    

14. L. G. Reyna and M. J. Ward, Meatball internal layer dynamics for the viscous Cahn-Hilliard equation, Methods and Applications of Analysis, 2 (1995): 285-306.    

15. C. C. Liu and J. X. Yin, Some properties of solutions for viscous Cahn-Hilliard equation, Northeast Math. J., 14 (1998): 455-466.

16. M. Grinfeld and A. Novick-Cohen, The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Amer. Math. Soc., 351 (1999): 2375-2406.

17. X. P. Zhao and C. C. Liu, Optimal control problem for viscous Cahn-Hilliard equation, Nonlinear Anal., 74 (2011): 6348-6357.    

18. A. B. Al'shin, M. O. Korpusov and A. G. Siveshnikov, Blow up in nonlinear Sobolev type equations, De Gruyter, 1 edition, Berlin, 2011.    

19. B. Hu, Blow-up theories for semilinear parabolic equations, Lect. Notes Math., Springer, Heidelberg, 2011.    

20. V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order parabolic, hyperbolic, idspersion and Schröinger equations, Monogr. Res. Notes Math., Chapman and Hall/CRC,2014.    

21. H. F. Di, Y. D. Shang and X. X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Cont. Dyn-B., 21 (2016): 781-801.    

22. C. C. Liu, Weak solutions for a class of metaparabolic equations, Appl. Anal., 87 (2008): 887-900.    

23. K. I. Khudaverdiyev and G. M. Farhadova, On global existence for generalized solution of one-dimensional non-self-adjoint mixed problem for a class of fourth order semilinear pseudoparabolic equations, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 31 (2009): 119-134.

24. H. J. Zhao and B. J. Xuan, Existence and convergence of solutions for the generalized BBMBurgers equations, Nonlinear Anal-Theor, 28 (1997): 1835-1849.    

25. G. A. Philippin, Blow-up phenomena for a class of fourth order parabolic problems, Proc. Amer. Math. Soc., 143 (2015): 2507-2513.    

26. G. A. Philippin and S. V. Piro, Behaviour in time of solutions to a class of fourth order evolution equations, J. Math. Anal. Appl., 436 (2016): 718-728.    

27. L. J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995): 136-155.    

28. R. B. Guenther and J. W. Lee, Partial differential equations of mathematical physics and integral equations, Prentice Hall, NJ, 1988.

29. G. W. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl., 351 (2009): 1-15.    

30. R. Z. Xu, S. Wang, Y. B. Yang and Y. H. Ding, Initial boundary value problem for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 92 (2013): 1403-1416.    

31. A. Khelghati and K. Baghaei, Blow-up phenomena for a class of fourth-order nonlinear wave equations with a viscous damping term, Math. Meth. Appl. Sci., in press, DOI: 10.1002/mma.3623, 2015.    

32. Z. J. Yang, Global existence asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term, J. Differ. Equations, 187 (2003): 520-540.    

33. L. E. Payne and D. H. Sattinger, Saddle points and instability on nonlinear hyperbolic equations, Israel Math. J., 22 (1975): 273-303.    

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