Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

In this paper, we investigate a initial value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace operator of order 0 < nu <= 1 and the nonlinear memory source term. For 0 < nu < 1, the problem will be considered on a bounded domain of Double-struck capital R-d. By some Sobolev embeddings and the properties of the Mittag-Leffler function, we will give some results on the existence and the uniqueness of mild solution for problem (1.1) below. When nu = 1, we will introduce some L-p - L-q estimates, and based on them we derive the global existence of a mild solution in the whole space Double-struck capital R-d.