Previous |  Up |  Next

Article

Keywords:
initial value problem for higher order parabolic equations; asymptotic behavior of solutions; critical exponent
Summary:
We consider the Cahn-Hilliard equation in $H^1(\mathbb R^N)$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $|u|\to \infty $ and logistic type nonlinearities. In both situations we prove the $H^2(\mathbb R^N)$-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).
References:
[1] Arrieta, J. M., Carvalho, A. N., Rodríguez-Bernal, A.: Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Differ. Equations 156 (1999), 376-406. DOI 10.1006/jdeq.1998.3612 | MR 1705387 | Zbl 0938.35077
[2] Arrieta, J. M., Cholewa, J. W., Dlotko, T., Rodríguez-Bernal, A.: Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56 (2004), 515-554. DOI 10.1016/j.na.2003.09.023 | MR 2035325 | Zbl 1058.35102
[3] Arrieta, J. M., Rodriguez-Bernal, A., Cholewa, J. W., Dlotko, T.: Linear parabolic equations in locally uniform spaces. Math. Models Methods Appl. Sci. 14 (2004), 253-293. DOI 10.1142/S0218202504003234 | MR 2040897 | Zbl 1058.35076
[4] Blömker, D., Maier-Paape, S., Wanner, T.: Spinodal decomposition for the Cahn-Hilliard-Cook equation. Commun. Math. Phys. 223 (2001), 553-582. DOI 10.1007/PL00005585 | MR 1866167 | Zbl 0993.60061
[5] Blömker, D., Maier-Paape, S., Wanner, T.: Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Trans. Am. Math. Soc. 360 (2008), 449-489. DOI 10.1090/S0002-9947-07-04387-5 | MR 2342011 | Zbl 1130.60066
[6] Bonfoh, A.: Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain. Q. Appl. Math. 64 (2006), 93-104. MR 2211379 | Zbl 1115.35025
[7] Bricmont, J., Kupiainen, A., Taskinen, J.: Stability of Cahn-Hilliard fronts. Commun. Pure Appl. Math. 52 (1999), 839-871. DOI 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I | MR 1682804 | Zbl 0939.35022
[8] Caffarelli, L. A., Muler, N. E.: An $L^\infty$ bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133 (1995), 129-144. DOI 10.1007/BF00376814 | MR 1367359
[9] Cahn, J. W., Hilliard, J. E.: Free energy of a nonuniform system. I: Interfacial free energy. J. Chem. Phys. 28 (1958), 258-267. DOI 10.1063/1.1744102
[10] Carvalho, A. N., Cholewa, J. W.: Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities. J. Math. Anal. Appl. 310 (2005), 557-578. DOI 10.1016/j.jmaa.2005.02.024 | MR 2022944 | Zbl 1077.35031
[11] Carvalho, A. N., Dlotko, T.: Dynamics of the viscous Cahn-Hilliard equation. J. Math. Anal. Appl. 344 (2008), 703-725. DOI 10.1016/j.jmaa.2008.03.020 | MR 2426301 | Zbl 1151.35008
[12] Cholewa, J. W., Dlotko, T.: Global attractor for the Cahn-Hilliard system. Bull. Aust. Math. Soc. 49 (1994), 277-292. DOI 10.1017/S0004972700016348 | MR 1265364 | Zbl 0803.35013
[13] Cholewa, J. W., Dlotko, T.: Global Attractors in Abstract Parabolic Problems. London Mathematical Society Lecture Note Series 278 Cambridge University Press, Cambridge (2000). MR 1778284 | Zbl 0954.35002
[14] Cholewa, J. W., Rodriguez-Bernal, A.: Dissipative mechanism of a semilinear higher order parabolic equation in $\mathbb{R}^N$. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 3510-3530. DOI 10.1016/j.na.2012.01.011 | MR 2901334
[15] Cholewa, J. W., Rodriguez-Bernal, A.: On the Cahn-Hilliard equation in $H^1(\mathbb{R}^N)$. J. Differ. Equations 253 (2012), 3678-3726. DOI 10.1016/j.jde.2012.08.033 | MR 2981268
[16] Cholewa, J. W., Rodriguez-Bernal, A.: Critical and supercritical higher order parabolic problems in $\mathbb{R}^N$. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 104 (2014), 50-74. DOI 10.1016/j.na.2014.03.013 | MR 3196888 | Zbl 1288.35281
[17] Dlotko, T., Kania, M. B., Sun, C.: Analysis of the viscous Cahn-Hilliard equation in $\mathbb{R}^N$. J. Differ. Equations 252 (2012), 2771-2791. DOI 10.1016/j.jde.2011.08.052 | MR 2860640
[18] Duan, L., Liu, S., Zhao, H.: A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation. J. Math. Anal. Appl. 372 (2010), 666-678. DOI 10.1016/j.jmaa.2010.06.009 | MR 2678892 | Zbl 1203.35040
[19] Elliott, C. M., Stuart, A. M.: Viscous Cahn-Hilliard equation II: Analysis. J. Differ. Equations 128 (1996), 387-414. MR 1398327 | Zbl 0855.35067
[20] Eyre, D. J.: Systems of Cahn-Hilliard Equations. University of Minnesota, AHPCRC Preprint 92-102, 1992. MR 1247174 | Zbl 0853.73060
[21] Grasselli, M., Schimperna, G., Zelik, S.: Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term. Nonlinearity 23 (2010), 707-737. DOI 10.1088/0951-7715/23/3/016 | MR 2593916 | Zbl 1198.35038
[22] Hale, J. K.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs 25 American Mathematical Society, Providence (1988). MR 0941371 | Zbl 0642.58013
[23] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 Springer, Berlin (1981). MR 0610244 | Zbl 0456.35001
[24] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58 (1975), 181-205. DOI 10.1007/BF00280740 | MR 0390516 | Zbl 0343.35056
[25] Korvola, T., Kupiainen, A., Taskinen, J.: Anomalous scaling for three-dimensional Cahn-Hilliard fronts. Commun. Pure Appl. Math. 58 (2005), 1077-1115. DOI 10.1002/cpa.20055 | MR 2143527 | Zbl 1078.35049
[26] Li, D., Zhong, C.: Global attractor for the Cahn-Hilliard system with fast growing nonlinearity. J. Differ. Equations 149 (1998), 191-210. DOI 10.1006/jdeq.1998.3429 | MR 1646238 | Zbl 0912.35029
[27] Liu, S., Wang, F., Zhao, H.: Global existence and asymptotics of solutions of the Cahn-Hilliard equation. J. Differ. Equations 238 (2007), 426-469. DOI 10.1016/j.jde.2007.02.014 | MR 2341432 | Zbl 1120.35044
[28] Miranville, A.: Long-time behavior of some models of Cahn-Hilliard equations in deformable continua. Nonlinear Anal., Real World Appl. 2 (2001), 273-304. MR 1835609 | Zbl 0989.35066
[29] Miranville, A.: Asymptotic behavior of the Cahn-Hilliard-Oono equation. J. Appl. Anal. Comput. 1 (2011), 523-536. MR 2889956 | Zbl 1304.35342
[30] Novick-Cohen, A.: On the viscous Cahn-Hilliard equation. Material instabilities in continuum mechanics. Proc. Symp., Heriot-Watt University, Edinburgh, 1985/86 J. M. Ball Oxford Science Publications Clarendon Press, Oxford 329-342 (1988). MR 0970531 | Zbl 0632.76119
[31] Novick-Cohen, A.: The Cahn-Hilliard equation. Handbook of Differential Equations: Evolutionary Equations IV Elsevier/North-Holland, Amsterdam (2008), 201-228. MR 2508166 | Zbl 1185.35001
[32] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences 68 Springer, New York (1988). DOI 10.1007/978-1-4684-0313-8 | MR 0953967 | Zbl 0662.35001
[33] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18 North-Holland Publishing Company, Amsterdam (1978). MR 0503903 | Zbl 0387.46033
[34] Zelik, S., Pennant, J.: Global well-posedness in uniformly local spaces for the Cahn-Hilliard equations in $\mathbb{R}^3$. Commun. Pure Appl. Anal. 12 (2013), 461-480. MR 2972440
Partner of
EuDML logo