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

Abstract

Abstract 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 $$u \in L^{2p} (\Omega ; \dot{H}_{0}^{1} (\mathcal {D})), \, p \ge 1$$ u ∈ L 2 p ( Ω ; H ˙ 0 1 ( D ) ) , p ≥ 1 and that the noise satisfies $$\Vert A^{\frac{\beta -1}{2}} \Vert _{\mathcal {L}_{2}^{0}} < \infty $$ ‖ A β - 1 2 ‖ L 2 0 < ∞ for $$\beta \in [0, 2]$$ 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 $$u \in L^{2p}(\Omega ; L^{6}(\mathcal {D})), \, p \ge 1$$ u ∈ L 2 p ( Ω ; L 6 ( D ) ) , p ≥ 1 . Under the more natural assumptions on the noise, $$\Vert A^{\frac{\sigma -\kappa }{2}} \Vert _{\mathcal {L}_{2}^{0}} < \infty $$ ‖ A σ - κ 2 ‖ L 2 0 < ∞ , where $$\sigma \in [\frac{d}{3}, 2]$$ σ ∈ [ d 3 , 2 ] ,