Czechoslovak Mathematical Journal, first online, pp. 1-23


Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík

Received March 28, 2019.   Published online August 18, 2020.

Abstract:  We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space $\Rd$ driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted $L^2$-space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.
Keywords:  reaction-diffusion equation; random attractor; spatially homogeneous noise
Classification MSC:  35B41, 60H15, 37L55, 35K57
DOI:  10.21136/CMJ.2020.0144-19

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] L. Arnold: Random Dynamical Systems. Springer Monographs in Mathematics, Springer, Berlin (1998). DOI 10.1007/978-3-662-12878-7 | MR 1723992 | Zbl 0906.34001
[2] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa, T. Dlotko: Linear parabolic equations in locally uniform spaces. Math. Models Methods Appl. Sci. 14 (2004), 253-293. DOI 10.1142/S0218202504003234 | MR 2040897 | Zbl 1058.35076
[3] P. W. Bates, H. Lisei, K. Lu: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6 (2006), 1-21. DOI 10.1142/S0219493706001621 | MR 2210679 | Zbl 1105.60041
[4] P. W. Bates, K. Lu, B. Wang: Tempered random attractors for parabolic equations in weighted spaces. J. Math. Phys. 54 (2013), Article ID 081505, 26 pages. DOI 10.1063/1.4817597 | MR 3135449 | Zbl 1288.35097
[5] Z. Brzeźniak: On Sobolev and Besov spaces regularity of Brownian paths. Stochastics Stochastics Rep. 56 (1996), 1-15. DOI 10.1080/17442509608834032 | MR 1396751 | Zbl 0890.60077
[6] Z. Brzeźniak: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 (1997), 245-295. DOI 10.1080/17442509708834122 | MR 1488138 | Zbl 0891.60056
[7] Z. Brzeźniak, Y. Li: Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains. Trans. Am. Math. Soc. 358 (2006), 5587-5629. DOI 10.1090/S0002-9947-06-03923-7 | MR 2238928 | Zbl 1113.60062
[8] Z. Brzeźniak, S. Peszat: Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process. Stud. Math. 137 (1999), 261-299. DOI 10.4064/sm-137-3-261-299 | MR 1736012 | Zbl 0944.60075
[9] Z. Brzeźniak, J. van Neerven: Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise. J. Math. Kyoto Univ. 43 (2003), 261-303. DOI 10.1215/kjm/1250283728 | MR 2051026 | Zbl 1056.60057
[10] T. Caraballo, J. A. Langa, V. S. Melnik, J. Valero: Pullback attractors of nonautonomous and stochastic multivalued dynamical systems. Set-Valued Anal. 11 (2003), 153-201. DOI 10.1023/A:1022902802385 | MR 1966698 | Zbl 1018.37048
[11] J. W. Cholewa, T. Dlotko: Cauchy problems in weighted Lebesgue spaces. Czech. Math. J. 54 (2004), 991-1013. DOI 10.1007/s10587-004-6447-z | MR 2099352 | Zbl 1080.35033
[12] H. Crauel, A. Debussche, F. Flandoli: Random attractors. J. Dyn. Differ. Equations 9 (1997), 307-341. DOI 10.1007/BF02219225 | MR 1451294 | Zbl 0884.58064
[13] H. Crauel, F. Flandoli: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100 (1994), 365-393. DOI 10.1007/BF01193705 | MR 1305587 | Zbl 0819.58023
[14] G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 152, Cambridge University Press, Cambridge (2014). DOI 10.1017/CBO9781107295513 | MR 3236753 | Zbl 1317.60077
[15] D. A. Dawson, H. Salehi: Spatially homogeneous random evolutions. J. Multivariate Anal. 10 (1980), 141-180. DOI 10.1016/0047-259X(80)90012-3 | MR 575923 | Zbl 0439.60051
[16] E. Feireisl: Bounded, locally compact global attractors for semilinear damped wave equations on ${\bf R}^N$. Differ. Integral Equ. 9 (1996), 1147-1156. MR 1392099 | Zbl 0858.35084
[17] F. Flandoli, B. Schmalfuss: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stochastics Stochastics Rep. 59 (1996), 21-45. DOI 10.1080/17442509608834083 | MR 1427258 | Zbl 0870.60057
[18] I. M. Gel'fand, N. Y. Vilenkin: Generalized Functions. Vol. 4. Applications of Harmonic Analysis. Academic Press, New York (1964). MR 0435834 | Zbl 0136.11201
[19] M. Grasselli, D. Pražák, G. Schimperna: Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories. J. Differ. Equations 249 (2010), 2287-2315. DOI 10.1016/j.jde.2010.06.001 | MR 2718659 | Zbl 1207.35068
[20] D. Haroske, H. Triebel: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators. I. Math. Nachr. 167 (1994), 131-156. DOI 10.1002/mana.19941670107 | MR 1285311 | Zbl 0829.46019
[21] J. Málek, D. Pražák: Large time behavior via the method of $l$-trajectories. J. Differ. Equations 181 (2002), 243-279. DOI 10.1006/jdeq.2001.4087 | MR 1907143 | Zbl 1187.37113
[22] M. Ondreját: Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. J. Evol. Equ. 4 (2004), 169-191. DOI 10.1007/s00028-003-0130-y | MR 2059301 | Zbl 1054.60068
[23] S. Peszat, J. Zabczyk: Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Processes Appl. 72 (1997), 187-204. DOI 10.1016/S0304-4149(97)00089-6 | MR 1486552 | Zbl 0943.60048
[24] S. Peszat, J. Zabczyk: Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116 (2000), 421-443. DOI 10.1007/s004400050257 | MR 1749283 | Zbl 0959.60044
[25] G. Tessitore, J. Zabczyk: Invariant measures for stochastic heat equations. Probab. Math. Stat. 18 (1998), 271-287. MR 1671596 | Zbl 0986.60057
[26] J. M. A. M. van Neerven, M. C. Veraar, L. Weis: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255 (2008), 940-993. DOI 10.1016/j.jfa.2008.03.015 | MR 2433958 | Zbl 1149.60039
[27] B. Wang: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equations 253 (2012), 1544-1583. DOI 10.1016/j.jde.2012.05.015 | MR 2927390 | Zbl 1252.35081
[28] S. V. Zelik: The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $\epsilon$-entropy. Math. Nachr. 232 (2001), 129-179. DOI 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K | MR 1871475 | Zbl 0989.35032

Affiliations:   Jakub Slavík, Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 00 Praha 8, Czech Republic, e-mail: slavik@utia.cas.cz


 
PDF available at: