Global existence and finite time blow-up for a class of thin-film equation

Article
First Online:
Received:
Revised:

Abstract

This paper deals with a class of thin-film equation, which was considered in Li et al. (Nonlinear Anal Theory Methods Appl 147:96–109, 2016), where the case of lower initial energy (\(J(u_0)\le d\) and d is a positive constant) was discussed, and the conditions on global existence or blow-up are given. We extend the results of this paper on two aspects: Firstly, we consider the upper and lower bounds of blow-up time and asymptotic behavior when \(J(u_0)<d\); secondly, we study the conditions on global existence or blow-up when \(J(u_0)>d\).

Keywords

Thin-film equation Potential wells Global existence Blow-up 

Mathematics Subject Classification

35B40 35K58 35K35 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Bian, S., Chen, L.: A nonlocal reaction diffusion equation and its relation with fujita exponent. J. Math. Anal. Appl. 444(2), 1479–1489 (2016)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bian, S., Chen, L., Latos, E.A.: Global existence and asymptotic behavior of solutions to a nonlocal Fisher–KPP type problem. Nonlinear Anal. 149, 165–176 (2015)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Budd, C.J., Williams, J.F.: Self-similar blow-up in higher-order semilinear parabolic equations. SIAM J. Appl. Math. 64(5), 1775–1809 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chapman, C.J., Proctor, M.R.E.: Nonlinear Rayleigh–Benard convection between poorly conducting boundaries. J. Fluid Mech. 101(4), 759–782 (1980)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cheung, K.L., Zhang, Z.Y.: Nonexistence of global solutions for a family of nonlocal or higher-order parabolic problems. Differ. Integral Equ. 25(7/8), 787–800 (2012)MathSciNetMATHGoogle Scholar
  7. 7.
    Evans, J.D., Galaktionov, V.A., Williams, J.F.: Blow-up and global asymptotics of the limit unstable Cahn–Hilliard equation. SIAM J. Math. Anal. 38(1), 64–102 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Evans, L.: Partial Differential Equations, 2nd edn. Wadsworth Brooks/Cole Mathematics, Pacific Grove (2010)MATHGoogle Scholar
  9. 9.
    Galaktionov, V.A.: Five types of blow-up in a semilinear fourth-order reactiondiffusion equation: an analytic numerical approach. Nonlinearity 22(7), 1695–1741 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gao, W.J., Han, Y.Z.: Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett. 24(5), 784–788 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gazzola, F., Weth, T.: Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ. Integral Equ. 18(9), 961–990 (2005)MathSciNetMATHGoogle Scholar
  12. 12.
    Jazar, M., Kiwan, R.: Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. I. Poincaré-AN 25(2), 215–218 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Khelghati, A., Baghaei, K.: Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy. Comp. Math. Appl. 70(5), 896–902 (2015)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    King, B.B., Stein, O., Winkler, M.: A fourth-order parabolic equation modeling epitaxial thin film growth. J. Math. Anal. Appl. 286(2), 459–490 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Li, Q.W., Gao, W.J., Han, Y.Z.: Global existence blow up and extinction for a class of thin-film equation. Nonlinear Anal. 147, 96–109 (2016)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Liu, Y.: Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition. Comput. Math. Appl. 66(10), 2092–2095 (2013)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Liu, Y.: Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions. Math. Comput. Model. 57(3–4), 926–931 (2013)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Messaoudi, S.A.: Global existence and decay of solutions to a system of Petrovsky. Math. Sci. Res. J. 6(11), 534–541 (2002)MathSciNetMATHGoogle Scholar
  19. 19.
    Messaoudi, S.A.: Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. 265(2), 296–308 (2002)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Niculescu, C.P., Rovena, I.: Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem. Nonlinear Anal. 75(1), 270–277 (2012)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Payne, L.E., Philippin, G.: Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions. Proc. Am. Math. Soc. 142(3), 625–631 (2013)MathSciNetMATHGoogle Scholar
  22. 22.
    Payne, L.E., Schaefer, P.W.: Bounds for blow-up time for the heat equation under nonlinear boundary conditions. Proc. R. Soc. Edinb. 139(6), 1289–1296 (2009)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Philippin, G.A.: Lower bounds for blow-up time in a class of nonlinear wave equations. Z. Angew. Math. Phys. 66(1), 129–134 (2015)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Qu, C.Y., Bai, X.L., Zheng, S.N.: Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions. J. Math. Anal. Appl. 412(1), 326–333 (2014)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Qu, C.Y., Liang, B.: Blow-up in a slow diffusive \(p\)-Laplace equation with the Neumann boundary conditions. Abstr. Appl. Anal. 2013(3), 551–552 (2013)MathSciNetGoogle Scholar
  26. 26.
    Qu, C.Y., Zhou, W.S.: Blow-up and extinction for a thin-film equation with initial-boundary value conditions. J. Math. Anal. Appl. 436(2), 796–809 (2015)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Song, J.C.: Lower bounds for the blow-up time in a non-local reaction–diffusion problem. Appl. Math. Lett. 24(5), 793–796 (2011)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Soufi, A., Jazar, M., Monneau, R.: A \(\gamma \)-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. I. Poincaré-AN 24(1), 17–39 (2007)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wang, X.L., Tian, F.Z., Li, G.: Nonlocal parabolic equation with conserved spatial integral. Arch. Math. 105(1), 93–100 (2015)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Wang, X.L., Wo, W.F.: Long time behavior of solutions for a scalar nonlocal reaction–diffusion equation. Arch. Math. 96(5), 483–490 (2011)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Xu, R.Z., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. 264(12), 2732–2763 (2013)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Zangwill, A.: Some causes and a consequence of epitaxial roughening. J. Cryst. Growth 163(1), 8–21 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhou, J.: Lower bounds for blow-up time of two nonlinear wave equations. Appl. Math. Lett. 45, 64–68 (2015)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Zhou, J.: Blow-up for a thin-film equation with positive initial energy. J. Math. Anal. Appl. 446(1), 1133–1138 (2016)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China

Personalised recommendations