Decay Rates and Initial Values for Time-Fractional Diffusion-Wave Equations

We consider a solution u(·,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α ∈ (0,2)\ {1} where t is a time variable. We first prove that a suitable norm of u(·,t) is bounded by the rate t −α for 0 α 1 and t 1−α for 1 α 2 for all large t > 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α = 1, the decay rate can keep some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.