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Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. (English) Zbl 1188.37076

The authors introduce the notion of quasi-continuity for a dynamical system \(\varphi\) on a Banach space \(B\) to mean that for any sequence \((t_n,x_n)\) converging to \((t,x)\) in \([0,\infty)\times B\), for which \(\bigl(\varphi(t_n)x_n\bigr)_{n\in\mathbb N}\) is bounded, \(\varphi(t_n)x_n\) converges weakly to \(\varphi(t)x\). This property is weaker than norm-to-weak continuity. The authors invoke this notion for random dynamical systems on a separable \(B\) in order to derive a sufficient condition for the existence of a random attractor, using asymptotic compactness properties formulated in terms of the Kuratowski measure of non-compactness.
The result is applied to stochastic reaction-diffusion equations on \(D\subset\mathbb R^d\) bounded, with a finite-dimensional additive Wiener process taking values in \(L^\infty(D)\). The nonlinearity of the drift being polynomially bounded, one has to consider solutions in \(L^p(D)\) for suitable \(p>2\). The induced random dynamical system is quasi-continuous, but it fails to be norm-to-weak continuous. It is shown to have a finite-dimensional random attractor \(A\) for bounded subsets of \(L^2(D)\), where attraction holds with respect to the \(L^p(D)\) norm for every \(p\geq2\). Finally, a comparison estimate for the fractal dimensions of \(A\) with respect to \(L^p(D)\) norms, \(p\geq2\), is given.

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
35B41 Attractors
35R60 PDEs with randomness, stochastic partial differential equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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