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Keywords:
reaction-diffusion equation; attractor; invariant measure; entropy; Poincaré-Bendixson theorem
Summary:
We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms.
References:
[1] Angenent, S.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390 (1988), 79-96. MR 0953678 | Zbl 0644.35050
[2] Eckmann, J.-P., Rougemont, J.: Coarsening by Ginzburg-Landau dynamics. Commun. Math. Phys. 199 (1998), 441-470. DOI 10.1007/s002200050508 | MR 1666859 | Zbl 1057.35508
[3] Fiedler, B., Mallet-Paret, J.: A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Ration. Mech. Anal. 107 (1989), 325-345. DOI 10.1007/BF00251553 | MR 1004714 | Zbl 0704.35070
[4] Gallay, T., Slijepčević, S.: Energy flow in formally gradient partial differential equations on unbounded domains. J. Dyn. Differ. Equations 13 (2001), 757-789. DOI 10.1023/A:1016624010828 | MR 1860285 | Zbl 1003.35085
[5] Gallay, T., Slijepčević, S.: Distribution of energy and convergence to equilibria in extended dissipative systems. (to appear) in J. Dyn. Differ. Equations.
[6] Joly, R., Raugel, G.: Generic Morse-Smale property for the parabolic equation on the circle. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 (2010), 1397-1440. DOI 10.1016/j.anihpc.2010.09.001 | MR 2738326 | Zbl 1213.35046
[7] Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications 54 Cambridge University Press, Cambridge (1995). MR 1326374 | Zbl 0878.58020
[8] Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of differential equations: Evolutionary Equations. Vol. IV C. M. Dafermos, M. Pokorný 103-200 Elsevier/North-Holland, Amsterdam (2008). DOI 10.1016/S1874-5717(08)00003-0 | MR 2508165 | Zbl 1221.37158
[9] Ollagnier, J. Moulin, Pinchon, D.: The variational principle. Studia Math. 72 (1982), 151-159. MR 0665415
[10] Slijepčević, S.: Extended gradient systems: Dimension one. Discrete Contin. Dyn. Syst. 6 (2000), 503-518. DOI 10.3934/dcds.2000.6.503 | MR 1757384 | Zbl 1009.37004
[11] Slijepčević, S.: The energy flow of discrete extended gradient systems. Nonlinearity 26 (2013), 2051-2079. DOI 10.1088/0951-7715/26/7/2051 | MR 3078107 | Zbl 1309.37075
[12] Slijepčević, S.: Ergodic Poincaré-Bendixson theorem for scalar reaction-diffusion equations. Preprint.
[13] Slijepčević, S.: The Aubry-Mather theorem for driven generalized elastic chains. Discrete Contin. Dyn. Syst. 34 (2014), 2983-3011. DOI 10.3934/dcds.2014.34.2983 | MR 3177671 | Zbl 1293.34017
[14] Turaev, D., Zelik, S.: Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete Contin. Dyn. Syst. 28 (2010), 1713-1751. DOI 10.3934/dcds.2010.28.1713 | MR 2679729 | Zbl 1213.35376
[15] Zelik, S.: Formally gradient reaction-diffusion systems in $\mathbb{R}^{n}$ have zero spatio-temporal topological entropy. Discrete Contin. Dyn. Syst. suppl. vol. (2003), 960-966. MR 2018206
[16] Zelik, S., Mielke, A.: Multi-pulse evolution and space-time chaos in dissipative systems. Mem. Am. Math. Soc. 198 (2009), 1-97. MR 2499464 | Zbl 1163.37003
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