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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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