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

We study the nonlinear stochastic time-fractional diffusion equation in the spatial domain R driven by a locally Lipschitz source satisfying ( (t) D-0+(alpha) - partial derivative(2)/partial derivative x(2)) u (t, x) = I-t(gamma) (F(t, x, u)), where x epsilon R, alpha epsilon (0, 1],gamma >= 1-alpha, the source term is defined F(t, x, u) = f (t, x, u(t, x)) +rho(t, x, u(t, x)). W (t, x) and W is the multiplicative space-time white noise. We investigate the existence, uniqueness of amaximal random field solution. Moreover, weprove the stability of the solution with respect to perturbed fractional orders alpha,gamma and the initial condition.