zbMATH — the first resource for mathematics

A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation. (English) Zbl 1203.35040
The authors consider the optimal decay estimates on the solutions of the Cauchy problem of the Cahn-Hilliard equation. It is shown that such a Cauchy problem admits a unique global smooth solution provided that the smooth nonlinear function satisfies a local growth condition.
Reviewer: Jiaqi Mo (Wuhu)

35B40 Asymptotic behavior of solutions to PDEs
35K30 Initial value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
Full Text: DOI
[1] Adams, R.A., Sobolev space, (1975), Academic Press New York
[2] Bricmont, J.; Kupiainen, A.; Taskinen, J., Stability of Cahn-Hilliard fronts, Comm. pure appl. math., 52, 839-871, (1999) · Zbl 0939.35022
[3] Caffarelli, L.A.; Muler, N.E., An \(L^\infty\)-bound for solutions of the Cahn-Hilliard equation, Arch. ration. mech. anal., 133, 129-144, (1995) · Zbl 0851.35010
[4] Charles, M. Ellot; Zheng, S.M., On the Cahn-Hilliard equation, Arch. ration. mech. anal., 96, 339-357, (1986) · Zbl 0624.35048
[5] Ding, X.X.; Wang, J.H., Global solution for a semilinear parabolic system, Acta math. sci., 3, 397-414, (1983) · Zbl 0592.65056
[6] Duoandikoetxea, J.; Zuazua, E., Monents, asses de Dirac et decomposition de fonctions, C. R. acad. sci. Paris, ser. I, 315, 693-698, (1992) · Zbl 0755.45019
[7] Hoff, D.; Smoller, J.A., Solutions in the large for certain nonlinear parabolic systems, Ann. inst. H. Poincaré anal. non linéaire, 2, 213-235, (1985) · Zbl 0578.35044
[8] Hoff, D.; Smoller, J.A., Global existence for parabolic conservation laws, J. differential equations, 68, 210-220, (1987) · Zbl 0624.35044
[9] Jeffery, A.; Zhao, H.J.; Jeffery, A.; Zhao, H.J., Global existence and optimal temporal decay estimates for systems of parabolic conservation laws II: the multidimensional case, Appl. anal., J. math. anal. appl., 217, 1-2, 597-623, (1998) · Zbl 0894.35047
[10] Karch, G., Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia math., 143, 2, 175-179, (2000) · Zbl 0964.35022
[11] Liu, S.Q.; Wang, F.; Zhao, H.J., Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. differential equations, 238, 2, 426-469, (2007) · Zbl 1120.35044
[12] Schonbek, M.E., Decay of solutions to parabolic conservation laws, Comm. partial differential equations, 7, 449-473, (1980) · Zbl 0476.35012
[13] Schonbek, M.E., Uniform decay rate for parabolic conservation laws, Nonlinear anal., 10, 9, 943-956, (1986) · Zbl 0617.35060
[14] Schonbek, M.E.; Rajopadhye, S., Asymptotic behavior of solutions to the Korteweg-de Vries-Burgers system, Ann. inst. H. Poincaré anal. non linéaire, 12, 425-457, (1995) · Zbl 0836.35144
[15] Ukai, S.; Yang, T.; Zhao, H.J., Convergence rate for the compressible Navier-Stokes equations with external force, J. hyperbolic differ. equ., 3, 3, 561-574, (2006) · Zbl 1184.35251
[16] Zhao, H.J., Asymptotics of solutions of nonlinear parabolic equations, J. differential equations, 191, 544-594, (2003) · Zbl 1036.35064
[17] Zheng, S.M., Asymptotic behavior of solution to the Cahn-Hilliard equation, Appl. anal., 23, 165-184, (1986) · Zbl 0582.34070
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.