New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C([−r,T];Lp(Ω,Wl,q(D))), Cμ((−∞,T];Lp(Ω,Wl,q(D))), and the weighted space Fμɛ((−∞,T];Lp(Ω,Wl,q(D))), instead of usual ones C([−r,T];L2(Ω,H)), Cμ((−∞,T];L2(Ω,H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H=L2(D) and Wl,q(D), and some well-known fractional tools. In addition, we also study the Hölder continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β→0+.