# zbMATH — the first resource for mathematics

Global existence and asymptotics of solutions of the Cahn-Hilliard equation. (English) Zbl 1120.35044
The objective of the paper is to study the Cauchy problem associated to a Cahn-Hilliard equation in an arbitrary dimension which contains a smooth nonlinear term. The authors establish the existence and uniqueness of the global solution provided the initial data is sufficiently small in the $$L^1$$-norm. Then they obtain temporal decay estimates for the solution. Finally, under a suitable growth condition on the nonlinearity, the asymptotics of the solution are described.

##### MSC:
 35K30 Initial value problems for higher-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations
Full Text:
##### References:
 [1] Charles, M. Ellot; Zheng, S.M., On the cahn – hilliard equation, Arch. ration. mech. anal., 96, 339-357, (1986) · Zbl 0624.35048 [2] Ding, X.X.; Wang, J.H., Global solution for a semilinear parabolic system, Acta math. sci., 3, 397-414, (1983) · Zbl 0592.65056 [3] Bricmont, J.; Kupiainen, A.; Taskinen, J., Stability of cahn – hilliard fronts, Comm. pure appl. math., 52, 839-871, (1999) · Zbl 0939.35022 [4] 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 [5] Jeffery, A.; Zhao, H.J., Global existence and optimal temporal decay estimates for systems of parabolic conservation laws I: the one-dimensional case, Appl. anal., J. math. anal. appl., 217, 1-2, 597-623, (1998), II: The multidimensional case · Zbl 0894.35047 [6] Zhao, H.J., Solutions in the large for certain nonlinear parabolic systems in arbitrary spatial dimensions, Appl. anal., 59, 349-376, (1995) · Zbl 0844.35047 [7] Zhao, H.J., Asymptotics of solutions of nonlinear parabolic equations, J. differential equations, 191, 544-594, (2003) · Zbl 1036.35064 [8] Zheng, S.M., Asymptotic behavior of solution to the cahn – hilliard equation, Appl. anal., 23, 165-184, (1986) · Zbl 0582.34070 [9] Strauss, W.A., Decay and asymptotic for $$u_{t t} - \operatorname{\Delta} u = F(u)$$, J. funct. anal., 2, 409-457, (1968) · Zbl 0182.13602 [10] 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 [11] Hoff, D.; Smoller, J.A., Global existence for parabolic conservation laws, J. differential equations, 68, 210-220, (1987) · Zbl 0624.35044 [12] Ding, X.X.; Zhao, H.J., Global solution of a semilinear parabolic systems, Acta math. sci., 11, 254-266, (1991) · Zbl 0746.35019 [13] Adams, R.A., Sobolev space, (1975), Academic Press New York [14] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer-Verlag New York · Zbl 0691.35001 [15] Schonbek, M.E., Decay of solutions to parabolic conservation laws, Comm. partial differential equations, 7, 449-473, (1980) · Zbl 0476.35012 [16] Schonbek, M.E., Uniform decay rate for parabolic conservation laws, Nonlinear anal., 10, 9, 943-956, (1986) · Zbl 0617.35060 [17] 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 [18] Karch, G., Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia math., 143, 2, 175-197, (2000) · Zbl 0964.35022 [19] Duoandikoetxea, J.; Zuazua, E., Moments, asses de Dirac et decomposition de fonctions, C. R. acad. sci. Paris ser. I, 315, 693-698, (1992) · Zbl 0755.45019 [20] Zhao, H.J., Decay estimates for the solutions of some multidimensional nonlinear evolution equations, Comm. partial differential equations, 25, 377-422, (2000) · Zbl 0985.35070
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.