Error estimates for stochastic time-fractional Allen–Cahn equations with integrated additive noises: smooth and nonsmooth cases

In recent work, Al-Maskari and Karaa [2] considered error estimates for a fully discrete scheme to solve a stochastic time-fractional Allen–Cahn equation driven by additive integrated noise, assuming the existence of a solution u ∈ L²ᵖ (Ω ; Ḣ₀¹ (D)), \, p ≥ 1 u ∈ L 2 p ( Ω ; H ˙ 0 1 ( D ) ) , p ≥ 1 and that the noise satisfies Aβ -1/2 _L₂⁰ < ∞ ‖ A β - 1 2 ‖ L 2 0 < ∞ for β ∈ [0, 2] β ∈ [ 0 , 2 ] . In this paper, we extend the results of [2]. We first apply the Yosida approximation to show that the stochastic time-fractional Allen–Cahn equation driven by additive integrated noise admits a unique solution u ∈ L²ᵖ(Ω ; L⁶(D)), \, p ≥ 1 u ∈ L 2 p ( Ω ; L 6 ( D ) ) , p ≥ 1 . Under the more natural assumptions on the noise, Aσ -κ /2 _L₂⁰ < ∞ ‖ A σ - κ 2 ‖ L 2 0 < ∞ , where σ ∈ [d/3, 2] σ ∈ [ d 3 , 2 ] ,