Random Attractors for a Class of Stochastic Partial Differential
Random attractors for a class of stochastic partial differential equations driven by general additive noise ∗ Benjamin Gess a, Wei Liu ay, Michael R¨ockner a;b a: Fakult¨atf¨urMathematik, Universit¨atBielefeld, D-33501 Bielefeld, Germany b: Department of Mathematics and Statistics, Purdue University, West Lafayette, 47906 IN, USA Abstract The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction-diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time fractional Brownian Motion and space-time L´evynoise as admissible random perturbations. Moreover, cases where the attractor consists of a single point are considered and bounds for the speed of attraction are obtained. AMS Subject Classification: 35B41, 60H15, 37L30, 35B40 Keywords: Random attractor; L´evynoise; fractional Brownian Motion; stochastic evolution equations; porous media equations; p-Laplace equation; reaction-diffusion equations. 1 Introduction Since the foundational work in [16, 17, 44] the long time behaviour of several examples of SPDE perturbed by additive noise has been extensively investigated by means of proving the existence of a global random attractor (cf. e.g. [8, 10, 11, 12, 19, 20, 31, 46, 47]). However, these results address only some specific examples of SPDE of semilinear type. To the best of our knowledge the only result concerning a non-semilinear SPDE, namely stochastic generalized porous media equations is given in [9]. In this work we provide a ∗Supported in part by DFG{Internationales Graduiertenkolleg \Stochastics and Real World Models", the SFB-701, the BiBoS-Research Center and NNSFC(10721091).
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