Random Attractors for Stochastic Partly Dissipative Systems

Abstract

We prove the existence of a global random attractor for a certain class of stochastic partly dissipative systems. These systems consist of a partial (PDE) and an ordinary differential equation (ODE), where both equations are coupled and perturbed by additive white noise. The deterministic counterpart of such systems and their long-time behaviour have already been considered but there is no theory that deals with the stochastic version of partly dissipative systems in their full generality. We also provide several examples for the application of the theory.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations

“…To show existence of solutions to (2.4), one could use the concept of mild solutions as employed for the FitzHugh-Nagumo SPDE with additive noise in [52]. The approach presented in [46], which we build upon, uses variational solutions [48,63,64] for equations with locally monotone coefficients [57,64].…”

“…[3,6,8,55,56,60]. Yet, the full SPDE variant (1.1) has only attracted major attention quite recently, including a large number of numerical studies [58,67,68,70,[73][74][75] as well as analytical studies regarding existence, regularity, invariant measures and attractors [4,7,9,52,54,77]. The question regarding stochastic stability of pulses for additive noise is far less studied but see [34] for multiplicative noise with a regularized equation (diffusion in the second variable) and see the review [51] for the stochastic Nagumo case (ε " 0).…”

Dynamics of Stochastic Reaction-Diffusion Equations

“…This is sufficient to obtain the necessary compactness results. Further details on this topic can be looked up in [125]. Before pointing out some concluding remarks we would like to emphasize that the structure of random attractors can be totally different from the deterministic case.…”

Travelling waves for discrete stochastic bistable equations

“…Using a localization and truncation argument, see e.g. [9,7], local-in-time results can be carried over polynomial nonlinearity f with one-sided Lipschitz condition such as in (SNagD), while globalin-time results have to exploit the sign in the cubic nonlinearity leading to dissipativity for large |u| [6,44,35]. Via monotone operator theory, one may see furthermore [42] that (SNagD) admits a variational solution in…”