In this article we present an Lp-theory (p ≥ 2) for the timefractional quasi-linear stochastic partial differential equations (SPDEs) of typewhere α ∈ (0, 2), β < α + 1 2 , and ∂ α t and ∂ β t denote the Caputo derivative of order α and β respectively. The processes w k t , k ∈ N = {1, 2, · · · }, are independent one-dimensional Wiener processes defined on a probability space Ω, L is a second order operator of either divergence or non-divergence type, and Λ k are linear operators of order up to two. The coefficients of the equations depend on ω(∈ Ω), t, x and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal Lp-regularity of the solutions. Our result certainly includes the Lp-theory for SPDEs driven by space-time white noise if the space dimension d < 4 − 2(2β − 1)α −1 . In particular, if β < 1/2 + α/4 then we can cover space-time white noise driven SPDEs with space dimension d = 1, 2, 3. This is a quite interesting result since in case of the classical SPDE, i.e. α = β = 1, d must be one.2010 Mathematics Subject Classification. 60H15, 45K05, 35R11.

We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the typet denotes the Caputo fractional derivative of order α, and Ω is a C 1 domain in R d . We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

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