Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.
@article{bwmeta1.element.bwnjournal-article-bcpv33z1p39bwm,
author = {Cholewa, Jan and D\l otko, Tomasz},
title = {Global solutions via partial information and the Cahn-Hilliard equation},
journal = {Banach Center Publications},
volume = {37},
year = {1996},
pages = {39-50},
zbl = {0852.35062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p39bwm}
}
Cholewa, Jan; Dłotko, Tomasz. Global solutions via partial information and the Cahn-Hilliard equation. Banach Center Publications, Tome 37 (1996) pp. 39-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p39bwm/
[000] [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[001] [2] H. Amann, Quasilinear evolution equations and parabolic systems, Trans. Amer. Math. Soc. 293 (1986), 191-227. | Zbl 0635.47056
[002] [3] J. W. Bebernes and K. Smitt, On the existence of maximal and minimal solutions for parabolic partial differential equations, Proc. Amer. Math. Soc. 73 (1979), 211-218. | Zbl 0399.35065
[003] [4] J. W. Cholewa, Classical Peano approach to quasilinear parabolic equations of arbitrary order, submitted.
[004] [5] J. W. Cholewa and T. Dłotko, Global attractor for the Cahn-Hilliard system, Bull. Austral. Math. Soc. 49 (1994), 277-293. | Zbl 0803.35013
[005] [6] T. Dłotko, Fourth order semilinear parabolic equations, Tsukuba J. Math. 16 (1992), 389-405. | Zbl 0798.35078
[006] [7] T. Dłotko, Global attractor for the Cahn-Hilliard equation in and , J. Differential Equations 113 (1994), 381-393. | Zbl 0828.35015
[007] [8] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. | Zbl 0224.35002
[008] [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, R.I., 1988. | Zbl 0642.58013
[009] [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981. | Zbl 0456.35001
[010] [11] W. Mlak, Hilbert Spaces and Operator Theory, Kluwer Academic Publishers and PWN, Dordrecht-Warszawa, 1991. | Zbl 0745.47001
[011] [12] J. L. Lions et E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. I, Dunod, Paris, 1968. | Zbl 0165.10801
[012] [13] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, R.I., 1968.
[013] [14] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors, Physica 16D (1985), 155-183. | Zbl 0592.35013
[014] [15] F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer, Berlin, 1984. | Zbl 0546.35003
[015] [16] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. | Zbl 0508.35002
[016] [17] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. | Zbl 0662.35001
[017] [18] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Deutscher Verlag Wiss., Berlin, 1978; also: North-Holland, Amsterdam, 1978. | Zbl 0387.46033
[018] [19] W. von Wahl, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985, A. Favini and E. Obrecht (eds.), Springer, Berlin, 1986, 254-266.