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 \(\left \{ \begin{array}{lr} u_t +{\rm div}(|u|^n\nabla\triangle u) = 0 \quad \textrm {in} \quad \Omega {\times}(0,T], \\ \frac {\partial u}{\partial\nu}=\frac{\partial}{\partial\nu}\triangle u = 0 \qquad \qquad \quad \textrm {on} \quad \partial\Omega{\times}(0,T],\\ u(x,0) = u_0(x) \qquad \qquad \quad \; \textrm {in} \quad \Omega. \end{array} \right. \)
Similar content being viewed by others
References
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
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
Bernis F., Friedman A. (1990). Higher order degenerate parabolic equations. J. Diff. Equ. 83: 179–206
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
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
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
Elliott C.M., Garcke H. (1996). On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27: 404–423
Grün G. (1995). Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding. Z. Anal. Anwendungen. 14: 541–574
Greenspan H.P. (1978). On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84: 24–51
Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)
Morrey, JR. C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York
Simon J. (1987). Compact sets in the space L p(0, T; B). Ann. Math. Pura Appl. 146: 65–96
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0113-3