A note on the Cahn-Hilliard equation in H 1 ( ℝ N ) involving critical exponent

Jan W. Cholewa; Aníbal Rodríguez-Bernal

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 269-283
  • ISSN: 0862-7959

Abstract

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We consider the Cahn-Hilliard equation in H 1 ( ℝ N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as | u | → ∞ and logistic type nonlinearities. In both situations we prove the H 2 ( 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).

How to cite

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Cholewa, Jan W., and Rodríguez-Bernal, Aníbal. "A note on the Cahn-Hilliard equation in $H^1(\mathbb {R}^N)$ involving critical exponent." Mathematica Bohemica 139.2 (2014): 269-283. <http://eudml.org/doc/261884>.

@article{Cholewa2014,
abstract = {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|\rightarrow \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).},
author = {Cholewa, Jan W., Rodríguez-Bernal, Aníbal},
journal = {Mathematica Bohemica},
keywords = {initial value problem for higher order parabolic equations; asymptotic behavior of solutions; critical exponent; initial value problem for higher order parabolic equations; asymptotic behavior of solutions; critical exponent},
language = {eng},
number = {2},
pages = {269-283},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the Cahn-Hilliard equation in $H^1(\mathbb \{R\}^N)$ involving critical exponent},
url = {http://eudml.org/doc/261884},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Cholewa, Jan W.
AU - Rodríguez-Bernal, Aníbal
TI - A note on the Cahn-Hilliard equation in $H^1(\mathbb {R}^N)$ involving critical exponent
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 269
EP - 283
AB - 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|\rightarrow \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).
LA - eng
KW - initial value problem for higher order parabolic equations; asymptotic behavior of solutions; critical exponent; initial value problem for higher order parabolic equations; asymptotic behavior of solutions; critical exponent
UR - http://eudml.org/doc/261884
ER -

References

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  1. Arrieta, J. M., Carvalho, A. N., Rodríguez-Bernal, A., 10.1006/jdeq.1998.3612, J. Differ. Equations 156 (1999), 376-406. (1999) Zbl0938.35077MR1705387DOI10.1006/jdeq.1998.3612
  2. Arrieta, J. M., Cholewa, J. W., Dlotko, T., Rodríguez-Bernal, A., 10.1016/j.na.2003.09.023, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56 (2004), 515-554. (2004) Zbl1058.35102MR2035325DOI10.1016/j.na.2003.09.023
  3. Arrieta, J. M., Rodriguez-Bernal, A., Cholewa, J. W., Dlotko, T., 10.1142/S0218202504003234, Math. Models Methods Appl. Sci. 14 (2004), 253-293. (2004) Zbl1058.35076MR2040897DOI10.1142/S0218202504003234
  4. Blömker, D., Maier-Paape, S., Wanner, T., 10.1007/PL00005585, Commun. Math. Phys. 223 (2001), 553-582. (2001) Zbl0993.60061MR1866167DOI10.1007/PL00005585
  5. Blömker, D., Maier-Paape, S., Wanner, T., 10.1090/S0002-9947-07-04387-5, Trans. Am. Math. Soc. 360 (2008), 449-489. (2008) Zbl1130.60066MR2342011DOI10.1090/S0002-9947-07-04387-5
  6. Bonfoh, A., 10.1090/S0033-569X-06-00988-3, Q. Appl. Math. 64 (2006), 93-104. (2006) Zbl1115.35025MR2211379DOI10.1090/S0033-569X-06-00988-3
  7. Bricmont, J., Kupiainen, A., Taskinen, J., 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I, Commun. Pure Appl. Math. 52 (1999), 839-871. (1999) Zbl0939.35022MR1682804DOI10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I
  8. Caffarelli, L. A., Muler, N. E., 10.1007/BF00376814, Arch. Ration. Mech. Anal. 133 (1995), 129-144. (1995) MR1367359DOI10.1007/BF00376814
  9. Cahn, J. W., Hilliard, J. E., 10.1063/1.1744102, J. Chem. Phys. 28 (1958), 258-267. (1958) DOI10.1063/1.1744102
  10. Carvalho, A. N., Cholewa, J. W., 10.1016/j.jmaa.2005.02.024, J. Math. Anal. Appl. 310 (2005), 557-578. (2005) Zbl1077.35031MR2022944DOI10.1016/j.jmaa.2005.02.024
  11. Carvalho, A. N., Dlotko, T., 10.1016/j.jmaa.2008.03.020, J. Math. Anal. Appl. 344 (2008), 703-725. (2008) Zbl1151.35008MR2426301DOI10.1016/j.jmaa.2008.03.020
  12. Cholewa, J. W., Dlotko, T., 10.1017/S0004972700016348, Bull. Aust. Math. Soc. 49 (1994), 277-292. (1994) Zbl0803.35013MR1265364DOI10.1017/S0004972700016348
  13. Cholewa, J. W., Dlotko, T., Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series 278 Cambridge University Press, Cambridge (2000). (2000) Zbl0954.35002MR1778284
  14. Cholewa, J. W., Rodriguez-Bernal, A., 10.1016/j.na.2012.01.011, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 3510-3530. (2012) MR2901334DOI10.1016/j.na.2012.01.011
  15. Cholewa, J. W., Rodriguez-Bernal, A., 10.1016/j.jde.2012.08.033, J. Differ. Equations 253 (2012), 3678-3726. (2012) MR2981268DOI10.1016/j.jde.2012.08.033
  16. Cholewa, J. W., Rodriguez-Bernal, A., 10.1016/j.na.2014.03.013, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 104 (2014), 50-74. (2014) Zbl1288.35281MR3196888DOI10.1016/j.na.2014.03.013
  17. Dlotko, T., Kania, M. B., Sun, C., 10.1016/j.jde.2011.08.052, J. Differ. Equations 252 (2012), 2771-2791. (2012) MR2860640DOI10.1016/j.jde.2011.08.052
  18. Duan, L., Liu, S., Zhao, H., 10.1016/j.jmaa.2010.06.009, J. Math. Anal. Appl. 372 (2010), 666-678. (2010) Zbl1203.35040MR2678892DOI10.1016/j.jmaa.2010.06.009
  19. Elliott, C. M., Stuart, A. M., Viscous Cahn-Hilliard equation II: Analysis, J. Differ. Equations 128 (1996), 387-414. (1996) Zbl0855.35067MR1398327
  20. Eyre, D. J., Systems of Cahn-Hilliard Equations, University of Minnesota, AHPCRC Preprint 92-102, 1992. Zbl0853.73060MR1247174
  21. Grasselli, M., Schimperna, G., Zelik, S., 10.1088/0951-7715/23/3/016, Nonlinearity 23 (2010), 707-737. (2010) Zbl1198.35038MR2593916DOI10.1088/0951-7715/23/3/016
  22. Hale, J. K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25 American Mathematical Society, Providence (1988). (1988) Zbl0642.58013MR0941371
  23. Henry, D., 10.1007/BFb0089647, Lecture Notes in Mathematics 840 Springer, Berlin (1981). (1981) Zbl0456.35001MR0610244DOI10.1007/BFb0089647
  24. Kato, T., 10.1007/BF00280740, Arch. Ration. Mech. Anal. 58 (1975), 181-205. (1975) Zbl0343.35056MR0390516DOI10.1007/BF00280740
  25. Korvola, T., Kupiainen, A., Taskinen, J., 10.1002/cpa.20055, Commun. Pure Appl. Math. 58 (2005), 1077-1115. (2005) Zbl1078.35049MR2143527DOI10.1002/cpa.20055
  26. Li, D., Zhong, C., 10.1006/jdeq.1998.3429, J. Differ. Equations 149 (1998), 191-210. (1998) Zbl0912.35029MR1646238DOI10.1006/jdeq.1998.3429
  27. Liu, S., Wang, F., Zhao, H., 10.1016/j.jde.2007.02.014, J. Differ. Equations 238 (2007), 426-469. (2007) Zbl1120.35044MR2341432DOI10.1016/j.jde.2007.02.014
  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. (2001) Zbl0989.35066MR1835609
  29. Miranville, A., Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput. 1 (2011), 523-536. (2011) Zbl1304.35342MR2889956
  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). (1988) Zbl0632.76119MR0970531
  31. Novick-Cohen, A., The Cahn-Hilliard equation, Handbook of Differential Equations: Evolutionary Equations IV Elsevier/North-Holland, Amsterdam (2008), 201-228. (2008) Zbl1185.35001MR2508166
  32. Temam, R., 10.1007/978-1-4684-0313-8, Applied Mathematical Sciences 68 Springer, New York (1988). (1988) Zbl0662.35001MR0953967DOI10.1007/978-1-4684-0313-8
  33. Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library 18 North-Holland Publishing Company, Amsterdam (1978). (1978) Zbl0387.46033MR0503903
  34. Zelik, S., Pennant, J., Global well-posedness in uniformly local spaces for the Cahn-Hilliard equations in ℝ 3 , Commun. Pure Appl. Anal. 12 (2013), 461-480. (2013) MR2972440

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