Advertisement

Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space

  • Jianzhong Zhang
  • Fushan LiEmail author
Article

Abstract

In this paper, we consider a nonlinear divergence form parabolic equation with time-dependent coefficient and inhomogeneous Neumann boundary condition. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution \(u(\pmb {x},t )\) exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space and constructing the unified functionals, we obtain upper and lower bounds of the blow-up time \(t^*\).

Keywords

Divergence form parabolic problems Inhomogeneous Neumann boundary condition Global existence Blow up Multidimensional space 

Mathematics Subject Classification

35K51 35K57 

Notes

Acknowledgements

The authors thank the editor and anonymous referees for their valuable suggestions and comments, which improved the presentation of this paper.

References

  1. 1.
    Bandle, C., Brunner, H.: Blowup in diffusion equations: a survey. J. Comput. Appl. Math. 97, 3–22 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ding, J., Hu, H.: Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions. J. Math. Anal. Appl. 433, 1718–1735 (2016) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ding, J., Shen, X.: Blow-up analysis for a class of nonlinear reaction diffusion equations with Robin boundary conditions. Math. Meth. Appl. Sci. 41, 1683–1696 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fang, Z., Wang, Y.: Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Z. Angew. Math. Phys. 66, 2525–2541 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Li, F., Li, J.: Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. J. Math. Anal. Appl. 385, 1005–1014 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Li, F., Li, J.: Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions. Bound. Value Probl. 2014, 219 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Liang, F.: Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux. Nonlinear Anal. 75, 2189–2198 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Payne, L.E., Schaefer, P.W.: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl. Anal. 85, 1301–1311 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Payne, L.E., Philippin, G.A., Vernier Piro, S.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. I. Z. Angew. Math. Phys. 61, 999–1007 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Payne, L.E., Philippin, G.A., Vernier Piro, S.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. II. Nonlinear Anal. 73, 971–978 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Payne, L.E., Philippin, G.A.: Blow-up in a class of non-linear parabolic problems with time-dependent coefficients under Robin type boundary conditions. Appl. Anal. 91, 2245–2256 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Payne, L.E., Philippin, G.A.: Blow-up phenomena parabolic problem under with time-dependent coefficients under Dirichlet type boundary conditions. Proc. Amer. Math. Soc. 141, 2309–2318 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Philippin, G.A.: Blow-up phenomena for a class of fourth-order parabolic problems. Proc. Amer. Math. Soc. 143, 2507–2513 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Quittner, R., Souplet, P.: Superlinear parabolic problems, in: Blow-up, Global Existence and Steady States, in: Birkhuser Advanced Texts, Birkhäuser, Basel, (2007)Google Scholar
  15. 15.
    Straughan, B.: Explosive Instabilities in Mechanics. Springer, Berlin (1998)CrossRefGoogle Scholar
  16. 16.
    Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in \(L^p\). Indiana Univ. Math. J. 29, 79–102 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Weissler, F.B.: Existence and nonexistence of global solutions for a heat equation. Israel J. Math. 38(1–2), 29–40 (1981)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuPeople’s Republic of China
  2. 2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

Personalised recommendations