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

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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.
  Article Metrics

Keywords metaparabolic equations; blow up; upper bound; lower bound

Citation: Huafei Di, Yadong Shang. Blow-up phenomena for a class of metaparabolic equations with time dependent coeffcient. AIMS Mathematics, 2017, 2(4): 647-657. doi: 10.3934/Math.2017.4.647


  • 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.    


This article has been cited by

  • 1. Jiali Yu, Yadong Shang, Huafei Di, On decay and blow-up of solutions for a nonlinear beam equation with double damping terms, Boundary Value Problems, 2018, 2018, 1, 10.1186/s13661-018-1067-y

Reader Comments

your name: *   your email: *  

Copyright Info: 2017, Huafei Di, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved