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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
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[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
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