The traditional wave equation models wave propagation in an ideal conducting medium. For characterizing the wave propagation in inhomogeneous media with frequency dependent power-law attenuation, the spacetime fractional wave equation appears; further incorporating the additive white Gaussian noise coming from many natural sources leads to the stochastic spacetime fractional wave equation. This paper discusses the Galerkin finite element approximations for the stochastic space-time fractional wave equation forced by an additive space-time white noise. We firstly discretize the space-time additive noise, which introduces a modeling error and results in a regularized stochastic space-time fractional wave equation; then the regularity of the regularized equation is analyzed. For the discretization in space, the finite element approximation is used and the definition of the discrete fractional Laplacian is introduced. We derive the mean-squared L 2 -norm priori estimates for the modeling error and for the approximation error to the solution of the regularized problem; and the numerical experiments are performed to confirm the estimates. For the time-stepping, we calculate the analytically obtained Mittag-Leffler type function.2000 Mathematics Subject Classification. Primary 26A33, 65L12, 65L20.

We present the existence of kernel sections (which are all compact, invariant and pullback attracting) of an infinite-dimensional general multi-valued process constructed by the set-valued backward extension of multi-valued semiprocesses. Moreover, the structure of the uniform attractors of a family of multi-valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi-valued processes are investigated. Finally, we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations.

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In this paper we study pullback attractors of reaction-diffusion equations on an unbounded domain with non-autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold. We first present the existence and structure of pullback attractors of multi-valued non-compact random dynamical systems. Then we prove the existence of pullback attractors in L 2 (R n ), L p (R n ) and H 1 (R n ) for multi-valued non-compact random dynamical systems associated with the reaction-diffusion equations on R n , and the identical relation of pullback attractors in different spaces is also provided. In particular, the measurability of pullback attractors is established for reaction-diffusion equations on R n .