Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term

<abstract><p>In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.</p></abstract>