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The dimension of attractors of nonautonomous partial differential equations. (English) Zbl 1047.35024
The paper provides an upper bound for the fractal dimension of pullback attractors of non-autonomous partial differential equations of the form \(u_t-\Delta u+f(t,u)=h(t)\) with Dirichlet boundary conditions on an open bounded set in \(\mathbb R^n\), under suitable conditions on \(f\) and \(h\). In particular, \(h\) is allowed to grow polynomially in time. The bound for the dimension of the attractor depends on the domain and on data of \(f\), but it can be chosen independent of time \(t\).

MSC:
35B41 Attractors
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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References:
[1] Metivier, J. Math. Pures Appl. 57 pp 133– (1978)
[2] Robinson, Infinite-dimensional dynamical systems (2002)
[3] Ladyzhenskaya, Zap. Nauchn. Sem. LOMI 182 pp 102– (1992)
[4] DOI: 10.1007/BF01193705 · Zbl 0819.58023
[5] DOI: 10.1007/BF02219225 · Zbl 0884.58064
[6] Constantin, Mem. Amer. Math. Soc. 53 (1985)
[7] Chepyzhov, Topol. Methods Nonlinear Anal. 7 pp 49– (1996)
[8] Chepyzhov, J. Math. Pures Appl. 73 pp 279– (1994)
[9] DOI: 10.1512/iumj.1993.42.42049 · Zbl 0819.35073
[10] Cheban, Nonlinear Dynam. Systems Theory 2 pp 9– (2002)
[11] Brezis, Anáalisis funcional (1984)
[12] DOI: 10.3934/dcds.1999.5.515 · Zbl 0963.37075
[13] DOI: 10.1016/S0167-6911(97)00107-2 · Zbl 0902.93043
[14] Hale, Asymptotic behavior of dissipative systems 25 (1988) · Zbl 0642.58013
[15] DOI: 10.1023/A:1021937715194 · Zbl 0931.35124
[16] Flandoli, Stochastics Stochastics Rep. 59 pp 21– (1996) · Zbl 0870.60057
[17] DOI: 10.1006/jmaa.1994.1251 · Zbl 0806.35074
[18] Eden, Exponential attractors for dissipative evolution equations 37 (1994) · Zbl 0842.58056
[19] Debussche, J. Math. Pures Appl. 77 pp 967– (1998) · Zbl 0919.58044
[20] DOI: 10.1080/07362999708809490 · Zbl 0888.60051
[21] DOI: 10.1007/BF01762184 · Zbl 0303.54016
[22] DOI: 10.1023/A:1022605313961 · Zbl 0927.37031
[23] DOI: 10.1023/A:1006769315268 · Zbl 0997.37058
[24] Temam, Infinite-dimensional dynamical systems in mechanics and physics (1988) · Zbl 0662.35001
[25] Schmalfu{\(\beta\)}, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999) pp 684– (2000)
[26] Schmalfu{\(\beta\)}, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour pp 185– (1992)
[27] DOI: 10.3934/dcds.1998.4.99 · Zbl 0954.37026
[28] DOI: 10.1080/00036818708839678 · Zbl 0609.35009
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