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

Abstract

We-24pt 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.