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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
,
Nguyen Dang Minh
,
Nguyen Nhu Lan
,
Nguyen Thi Mong Ngoc
Abstract
We study the nonlinear stochastic time-fractional diffusion equation in the spatial domain $\mathbb R$ driven by a locally Lipschitz source satisfying
$$\begin{aligned} \left( {~}_{t}D_{0^{+}}^{\alpha } - \frac{\partial ^{2} }{\partial x^{2}}\right) u(t,x) = I_{t}^{\gamma }\left( F(t,x,u)\right) , \end{aligned}$$
where $x\in \mathbb {R},\alpha \in (0,1],\gamma \ge 1-\alpha$, the source term is defined $F(t,x,u) = f(t,x,u(t,x))+ \rho (t,x,u(t,x))\dot{W}(t,x)$ 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 $\alpha , \gamma$ and the initial condition.