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The attractor for a nonlinear reaction-diffusion system in an unbounded domain. (English) Zbl 1041.35016
The authors study attractors of reaction-diffusion systems in unbounded domains including a nonlinear dependence on the gradient. Existence of solutions and a priori bounds are obtained in exponentially and algebraically weighted spaces under a sign condition for the nonlinearity and appropriate growth conditions on the gradient dependence. The lack of uniqueness is circumvened by the concept of trajectory attractors. Results include the existence of attractors on scales of algebraically weighted Sobolev spaces. Examples are provided where the attractor is infinite dimensional. For weights enforcing decay at infinity, the attractor is shown to be finite-dimensional.

MSC:
35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K57 Reaction-diffusion equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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[1] Abergel, J Differential Equations 83 pp 85– (1990) · Zbl 0706.35058
[2] Babin, Izv Ross Akad Nauk Ser Mat 58 pp 3– (1994)
[3] Babin, Mat Sb (NS) 126(68) pp 397– (1985)
[4] Babin, Proc Roy Soc Edinburgh Sect A 116 pp 221– (1990) · Zbl 0721.35029
[5] ; Attraktory èvolyutsionnykh uravneni??. [Attractors of evolution equations.] Nauka, Moscow, 1989; translated and revised from the 1989 Russian original by Babin. Studies in Mathematics and Its Applications, 25. North-Holland, Amsterdam, 1992.
[6] Chepyzhov, Comm Pure Appl Math 53 pp 647– (2000) · Zbl 1022.37048
[7] Chepyzhov, J Math Pures Appl (9) 73 pp 279– (1994)
[8] Chepyzhov, C R Acad Sci Paris Sér I Math 321 pp 153– (1995)
[9] Chepyzhov, C R Acad Sci Paris Sér I Math 321 pp 1309– (1995)
[10] Chepyzhov, J Math Pures Appl (9) 76 pp 913– (1997) · Zbl 0896.35032
[11] ; ; ; ; One-parameter semigroups. CWI Monographs, 5. North-Holland, Amsterdam?New York, 1987.
[12] Collet, Comm Math Phys 200 pp 699– (1999) · Zbl 0920.35071
[13] Crooks, Topol Methods Nonlinear Anal 11 pp 19– (1998) · Zbl 0920.35075
[14] ; Linear operators. Part I. General theory. Pure and Applied Mathematics, Vol. 7. Interscience, New York?London, 1958;
[15] ; Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space. Interscience. Wiley, New York?London, 1963.
[16] Long-time behaviour of dynamical systems: New concepts. Theory of nonlinear evolution equations and its applications. Proceedings of the symposium held at the Research Institute for Mathematical Sciences, Kyoto University, 1998. Kyoto University, Research Institute for Mathematical Sciences, Kyoto, 1998.
[17] Efendiev, Discrete Contin Dynam Systems 5 pp 399– (1999) · Zbl 0959.35025
[18] Feireisl, C R Acad Sci Paris Sér I Math 319 pp 147– (1994)
[19] Fiedler, J Math Pures Appl (9) 77 pp 879– (1998) · Zbl 0917.35011
[20] Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, R.I., 1988. · Zbl 0642.58013
[21] Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer, Berlin?New York, 1981.
[22] ; Applications of functional analysis and operator theory. Mathematics in Science and Engineering, 146. Academic [Harcourt Brace Jovanovich], New York?London, 1980.
[23] Topological methods in the theory of nonlinear integral equations. Macmillan, New York, 1964.
[24] ; ; Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, 23. American Mathematical Society, Providence, R.I., 1967.
[25] ; Inequalities for the moments of the eigenvalues of the Schrödinger equations and their relation to Sobolev inequalities, 269-303. Studies in Mathematical Physics: Essays in Honor of Valentine Bergmann. Princeton University, Princeton, 1976.
[26] ; Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, 17. Dunod, Paris, 1968.
[27] Merino, J Differential Equations 132 pp 87– (1996) · Zbl 0867.35045
[28] Mielke, Nonlinearity 10 pp 199– (1997) · Zbl 0905.35043
[29] Mielke, Phys D 117 pp 106– (1998) · Zbl 0939.35033
[30] Topics in nonlinear functional analysis. Notes by R. A. Artino. Lecture Notes, 1973-1974. Courant Institute of Mathematical Sciences, New York University, New York, 1974.
[31] Schulze, Rend Accad Naz Sci XL Mem Mat Appl (5) 23 pp 125– (1999)
[32] Some applications of functional analysis in mathematical physics. Third edition. Nauka, Moscow, 1988. Translations of Mathematical Monographs, 90. American Mathematical Society, Providence, R.I., 1991.
[33] Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, 68. Springer, New York?Berlin, 1988.
[34] Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library, 18. North-Holland, Amsterdam?New York, 1978.
[35] Vishik, Mat Sb 187 pp 21– (1996)
[36] ; Analysis in classes of discontinuous functions and equations of mathematical physics. Mechanics: Analysis, 8. Nijhoff, Dordrecht, 1985.
[37] Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer, Berlin, 1985; translated from the German by Peter R. Wadsack. Springer, New York?Berlin, 1986.
[38] Zelik, Mat Zametki 63 pp 135– (1998)
[39] Zelik, Mat Zametki 65 pp 941– (1999)
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