Abstract: This paper discusses the well-posedness problem of fractional nonclassical diffusion equations with time-dependent dissipation coefficients. Using the nonclassical Faedo-Galerkin method, the interpolation inequality and the control convergence principle, the existence, uniqueness and continuous dependence on initial values of the global weak solution in $\mathcal{H}^{\theta} (0 < \theta \leq 1)$ for the equations are obtained, where the nonlinearity $f$ satisfies the polynomial growth of arbitrary order.

Key words: Time-dependent diffusion coefficient; , Fractional nonclassical diffusion equation; , Global weak solution; , Polynomial growth of arbitrary order