| Title:
|
Attractors for stochastic reaction-diffusion equation with additive homogeneous noise (English) |
| Author:
|
Slavík, Jakub |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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71 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
21-43 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space $\mathbb{R}^d$ 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. (English) |
| Keyword:
|
reaction-diffusion equation |
| Keyword:
|
random attractor |
| Keyword:
|
spatially homogeneous noise |
| MSC:
|
35B41 |
| MSC:
|
35K57 |
| MSC:
|
37L55 |
| MSC:
|
60H15 |
| idZBL:
|
07332705 |
| idMR:
|
MR4226470 |
| DOI:
|
10.21136/CMJ.2020.0144-19 |
| . |
| Date available:
|
2021-03-12T16:09:15Z |
| Last updated:
|
2023-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148728 |
| . |
| Reference:
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[1] Arnold, L.: Random Dynamical Systems.Springer Monographs in Mathematics, Springer, Berlin (1998). Zbl 0906.34001, MR 1723992, 10.1007/978-3-662-12878-7 |
| Reference:
|
[2] Arrieta, J. M., Rodriguez-Bernal, A., Cholewa, J. W., Dlotko, T.: Linear parabolic equations in locally uniform spaces.Math. Models Methods Appl. Sci. 14 (2004), 253-293. Zbl 1058.35076, MR 2040897, 10.1142/S0218202504003234 |
| Reference:
|
[3] Bates, P. W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical systems.Stoch. Dyn. 6 (2006), 1-21. Zbl 1105.60041, MR 2210679, 10.1142/S0219493706001621 |
| Reference:
|
[4] Bates, P. W., Lu, K., Wang, B.: Tempered random attractors for parabolic equations in weighted spaces.J. Math. Phys. 54 (2013), Article ID 081505, 26 pages. Zbl 1288.35097, MR 3135449, 10.1063/1.4817597 |
| Reference:
|
[5] Brzeźniak, Z.: On Sobolev and Besov spaces regularity of Brownian paths.Stochastics Stochastics Rep. 56 (1996), 1-15. Zbl 0890.60077, MR 1396751, 10.1080/17442509608834032 |
| Reference:
|
[6] Brzeźniak, Z.: On stochastic convolution in Banach spaces and applications.Stochastics Stochastics Rep. 61 (1997), 245-295. Zbl 0891.60056, MR 1488138, 10.1080/17442509708834122 |
| Reference:
|
[7] Brzeźniak, Z., Li, Y.: Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains.Trans. Am. Math. Soc. 358 (2006), 5587-5629. Zbl 1113.60062, MR 2238928, 10.1090/S0002-9947-06-03923-7 |
| Reference:
|
[8] Brzeźniak, Z., Peszat, S.: Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process.Stud. Math. 137 (1999), 261-299. Zbl 0944.60075, MR 1736012, 10.4064/sm-137-3-261-299 |
| Reference:
|
[9] Brzeźniak, Z., Neerven, J. van: Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise.J. Math. Kyoto Univ. 43 (2003), 261-303. Zbl 1056.60057, MR 2051026, 10.1215/kjm/1250283728 |
| Reference:
|
[10] Caraballo, T., Langa, J. A., Melnik, V. S., Valero, J.: Pullback attractors of nonautonomous and stochastic multivalued dynamical systems.Set-Valued Anal. 11 (2003), 153-201. Zbl 1018.37048, MR 1966698, 10.1023/A:1022902802385 |
| Reference:
|
[11] Cholewa, J. W., Dlotko, T.: Cauchy problems in weighted Lebesgue spaces.Czech. Math. J. 54 (2004), 991-1013. Zbl 1080.35033, MR 2099352, 10.1007/s10587-004-6447-z |
| Reference:
|
[12] Crauel, H., Debussche, A., Flandoli, F.: Random attractors.J. Dyn. Differ. Equations 9 (1997), 307-341. Zbl 0884.58064, MR 1451294, 10.1007/BF02219225 |
| Reference:
|
[13] Crauel, H., Flandoli, F.: Attractors for random dynamical systems.Probab. Theory Relat. Fields 100 (1994), 365-393. Zbl 0819.58023, MR 1305587, 10.1007/BF01193705 |
| Reference:
|
[14] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions.Encyclopedia of Mathematics and Its Applications 152, Cambridge University Press, Cambridge (2014). Zbl 1317.60077, MR 3236753, 10.1017/CBO9781107295513 |
| Reference:
|
[15] Dawson, D. A., Salehi, H.: Spatially homogeneous random evolutions.J. Multivariate Anal. 10 (1980), 141-180. Zbl 0439.60051, MR 575923, 10.1016/0047-259X(80)90012-3 |
| Reference:
|
[16] Feireisl, E.: Bounded, locally compact global attractors for semilinear damped wave equations on {$\bold R^N$}.Differ. Integral Equ. 9 (1996), 1147-1156. Zbl 0858.35084, MR 1392099 |
| Reference:
|
[17] Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise.Stochastics Stochastics Rep. 59 (1996), 21-45. Zbl 0870.60057, MR 1427258, 10.1080/17442509608834083 |
| Reference:
|
[18] Gel'fand, I. M., Vilenkin, N. Y.: Generalized Functions. Vol. 4. Applications of Harmonic Analysis.Academic Press, New York (1964). Zbl 0136.11201, MR 0435834 |
| Reference:
|
[19] Grasselli, M., Pražák, D., Schimperna, G.: Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories.J. Differ. Equations 249 (2010), 2287-2315. Zbl 1207.35068, MR 2718659, 10.1016/j.jde.2010.06.001 |
| Reference:
|
[20] Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators. I.Math. Nachr. 167 (1994), 131-156. Zbl 0829.46019, MR 1285311, 10.1002/mana.19941670107 |
| Reference:
|
[21] Málek, J., Pražák, D.: Large time behavior via the method of $l$-trajectories.J. Differ. Equations 181 (2002), 243-279. Zbl 1187.37113, MR 1907143, 10.1006/jdeq.2001.4087 |
| Reference:
|
[22] Ondreját, M.: 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. Zbl 1054.60068, MR 2059301, 10.1007/s00028-003-0130-y |
| Reference:
|
[23] Peszat, S., Zabczyk, J.: Stochastic evolution equations with a spatially homogeneous Wiener process.Stochastic Processes Appl. 72 (1997), 187-204. Zbl 0943.60048, MR 1486552, 10.1016/S0304-4149(97)00089-6 |
| Reference:
|
[24] Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations.Probab. Theory Relat. Fields 116 (2000), 421-443. Zbl 0959.60044, MR 1749283, 10.1007/s004400050257 |
| Reference:
|
[25] Tessitore, G., Zabczyk, J.: Invariant measures for stochastic heat equations.Probab. Math. Stat. 18 (1998), 271-287. Zbl 0986.60057, MR 1671596 |
| Reference:
|
[26] Neerven, J. M. A. M. van, Veraar, M. C., Weis, L.: Stochastic evolution equations in UMD Banach spaces.J. Funct. Anal. 255 (2008), 940-993. Zbl 1149.60039, MR 2433958, 10.1016/j.jfa.2008.03.015 |
| Reference:
|
[27] Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems.J. Differ. Equations 253 (2012), 1544-1583. Zbl 1252.35081, MR 2927390, 10.1016/j.jde.2012.05.015 |
| Reference:
|
[28] Zelik, S. V.: The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $\epsilon$-entropy.Math. Nachr. 232 (2001), 129-179. Zbl 0989.35032, MR 1871475, 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K |
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