Skip to main content
Log in

A note on a fourth order degenerate parabolic equation in higher space dimensions

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions \(\left \{ \begin{array}{lr} u_t +{\rm div}(|u|^n\nabla\triangle u) = 0 \quad \textrm {in} \quad \Omega {\times}(0,T], \\ \frac {\partial u}{\partial\nu}=\frac{\partial}{\partial\nu}\triangle u = 0 \qquad \qquad \quad \textrm {on} \quad \partial\Omega{\times}(0,T],\\ u(x,0) = u_0(x) \qquad \qquad \quad \; \textrm {in} \quad \Omega. \end{array} \right. \)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beretta E., Bertsch M., Dal Passo R. (1995). Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 129: 175–200

    Article  MATH  Google Scholar 

  2. Bernis F. (1995). Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems. In: Diaz, J.I., Herrero, M.A., Linan, A., Vazquez, J.L. (eds) Free Boundary Problems: Theory and Applications. Pitman Research Notes in Mathematics, vol. 323, pp 40–56. Longman, Harlow

    Google Scholar 

  3. Bernis F., Friedman A. (1990). Higher order degenerate parabolic equations. J. Diff. Equ. 83: 179–206

    Article  MATH  Google Scholar 

  4. Bertozzi A.L., Pugh M. (1994). The lubrication approximation for thin viscous film: regularity and long time behaviour of weak solutions. Comm. Pure Appl. Math. 49: 85–123

    Article  Google Scholar 

  5. Bertsch M., Dal Passo R., Garcke H., Grün G. (1998). The thin viscous flow equation in higher space dimensions. Adv Differ. Equ. 3: 417–440

    MATH  Google Scholar 

  6. Dal Passo R., Garcke H., Grün G. (1998). On a fourth order degenerate parabolic equation: global entropy estimates, existence, and qualitative behaviour of solutions. SIAM J. Math. Anal. 29: 321–342

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Grün G. (1995). Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding. Z. Anal. Anwendungen. 14: 541–574

    MATH  Google Scholar 

  9. Greenspan H.P. (1978). On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84: 24–51

    Article  Google Scholar 

  10. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)

  11. Morrey, JR. C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York

  12. Simon J. (1987). Compact sets in the space L p(0, T; B). Ann. Math. Pura Appl. 146: 65–96

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Junjie Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J. A note on a fourth order degenerate parabolic equation in higher space dimensions. Math. Ann. 339, 251–285 (2007). https://doi.org/10.1007/s00208-007-0113-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-007-0113-3

Mathematics Subject Classification (2000)

Navigation