Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. (English) Zbl 1306.37091

Most of stochastic partial differential equations may possess a regular solution. For example, when initial data are located in a Banach space, called an initial space, the corresponding solution may belong to another Banach space, called a terminate space. To depict better the dynamics for such equations, the concept of a bi-spatial random attractor (BSRA) for a random dynamical system (RDS) is introduced in this paper.
A BSRA is a compact and invariant random set attracting some subsets of the initial space according to the topology of the terminate space. This concept generalizes some common concepts of random attractors and tempered random attractors.
The first goal of this paper is to establish an existence criterion of a BSRA for a RDS. The second goal is to establish a continuity criterion of a family of BSRA under the Hausdorff semi-distance of the terminate space as well as the initial space. The third goal is to apply the obtained theoretical result to stochastic semilinear Laplacian equations on an unbounded domain with multiplicative noises.
The authors also consider another continuity problem of attractors for a (single) stochastic PDE on an unbounded domain. For the stochastic semilinear Laplacian equation, they prove that the attractor on the entire space is just the closure of the union of attractors over all bounded domains.


37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B40 Asymptotic behavior of solutions to PDEs
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