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Volume 28, Issue 1
Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains

Zhi Wang & XianYun Du

DOI: 10.4208/jpde.v28.n1.6

J. Part. Diff. Eq., 28 (2015), pp. 47-73.

Published online: 2015-03

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  • Abstract
In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback random attractor in L²(R^n) × L²(R^n). We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.
  • Keywords

Reaction diffusion equation random dynamical systems random attractors asymptotic compactness Sobolev compact embedding

  • AMS Subject Headings

35R60, 37L55, 60H10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wang111zhi@126.com (Zhi Wang)

du2011@foxmail.com (XianYun Du)

  • BibTex
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  • TXT
@Article{JPDE-28-47, author = {Wang , Zhi and Du , XianYun}, title = {Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains}, journal = {Journal of Partial Differential Equations}, year = {2015}, volume = {28}, number = {1}, pages = {47--73}, abstract = { In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback random attractor in L²(R^n) × L²(R^n). We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v28.n1.6}, url = {http://global-sci.org/intro/article_detail/jpde/5102.html} }
TY - JOUR T1 - Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains AU - Wang , Zhi AU - Du , XianYun JO - Journal of Partial Differential Equations VL - 1 SP - 47 EP - 73 PY - 2015 DA - 2015/03 SN - 28 DO - http://doi.org/10.4208/jpde.v28.n1.6 UR - https://global-sci.org/intro/article_detail/jpde/5102.html KW - Reaction diffusion equation KW - random dynamical systems KW - random attractors KW - asymptotic compactness KW - Sobolev compact embedding AB - In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback random attractor in L²(R^n) × L²(R^n). We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.
Wang , Zhi and Du , XianYun. (2015). Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains. Journal of Partial Differential Equations. 28 (1). 47-73. doi:10.4208/jpde.v28.n1.6
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