Random attractors for degenerate stochastic partial differential equations

Abstract

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate p-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate p-Laplace equations we prove that the deterministic, infinite dimensional attractor collapses to a single random point if enough noise is added.

34 pages; The final publication is available at http://link.springer.com/article/10.1007%2Fs10884-013-9294-5

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• Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts
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• Weak Pullback Mean Random Attractors for Stochastic Evolution Equations and Applications
• Small Time Asymptotics for SPDEs with Locally Monotone Coefficients
• Ergodicity for singular-degenerate porous media equations
• Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$
• A random attractor for stochastic porous media equations on infinite lattices
• Finite speed of propagation for stochastic porous media equations
• Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations
• Asymptotic Dynamics of Stochastic $p$-Laplace Equations on Unbounded Domains