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Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources

Dang Duc Trong , $icon-email$ Nguyen Dang Minh , Nguyen Nhu Lan , Nguyen Thi Mong Ngoc

Abstract

We study the nonlinear stochastic time-fractional diffusion equation in the spatial domain $R$ driven by a locally Lipschitz source satisfying

$\begin{array}{r} (_{t} D_{0^{+}}^{α} - \frac{\partial^{2}}{\partial x^{2}}) u (t, x) = I_{t}^{γ} (F (t, x, u)), \end{array}$

where $x \in R, α \in (0, 1], γ \geq 1 - α$ , the source term is defined $F (t, x, u) = f (t, x, u (t, x)) + ρ (t, x, u (t, x)) \dot{W} (t, x)$ and $W$ is the multiplicative space-time white noise. We investigate the existence, uniqueness of a maximal random field solution. Moreover, we prove the stability of the solution with respect to perturbed fractional orders $α, γ$ and the initial condition.

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Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources

Abstract