Existence and blow-up of solutions of a fourth-order nonlinear diffusion equation. (English) Zbl 1156.35406
Summary: We are concerned with the existence, uniqueness and blow up of solutions for a fourth-order nonlinear diffusion equation.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
References:
| [1] | Bertozzi, A. L., Symmetric singularity formation in lubrication-type equations for interface motion, SIAM J. Appl. Math., 56, 681-714 (1996) · Zbl 0856.35002 |
| [2] | Bertozzi, A. L.; Pugh, M. C., Finite-time blow-up of solutions of some long-wave unstable thin-film equations, Indiana Univ. Math. J., 49, 1323-1366 (2000) · Zbl 0978.35007 |
| [3] | Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys., 28, 258-267 (1958) · Zbl 1431.35066 |
| [4] | Elliott, C. M.; Zheng, S. M., On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96, 339-357 (1986) · Zbl 0624.35048 |
| [5] | Jensen, O. E.; Chini, G. P.; King, J. R., Thin-film flows near isolated humps and interior corners, J. Eng. Math., 50, 289-309 (2004) · Zbl 1074.76008 |
| [6] | King, J. R., Two generalisations of the thin film equation, Math. Comput. Modelling, 34, 737-756 (2001) · Zbl 1001.35073 |
| [7] | Lidia, A.; Lorenzo, G., Doubly nonlinear thin-film equations in one space dimension, Arch. Ration. Mech. Anal., 173, 89-131 (2004) · Zbl 1064.76012 |
| [8] | Liu, C. C., Some properties of solutions for the generalized thin film equation in one space dimension, Bol. Asoc. Mat. Venez., 12, 43-52 (2005) · Zbl 1099.35115 |
| [9] | J.D. Murray, Mathematical Biology, Springer, Berlin, p. 767.; J.D. Murray, Mathematical Biology, Springer, Berlin, p. 767. · Zbl 0704.92001 |
| [10] | Timo, K.; Antti, K.; Jari, T., Anomalous scaling for three-dimensional Cahn-Hilliard fronts, Commun. Pure Appl. Math., 58, 1077-1115 (2005) · Zbl 1078.35049 |
| [11] | Wang, R. H., On the Schauder type theory about the general parabolic boundary value problem, Acta Scient. Natur. Univ. Jilinensis, 2, 35-64 (1964) |
| [12] | Yin, J. X., On the Cahn-Hilliard equation with nonlinear principal part, J. Partial Differential Equations, 7, 77-96 (1994) · Zbl 0823.35036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
