Existence and uniqueness of weak solutions for a class of fractional superdiffusion equations

Abstract

In this paper, we consider the existence and uniqueness of weak solutions for a class of fractional superdiffusion equations with initial-boundary conditions. For a multidimensional fractional drift superdiffusion equation, we just consider the simplest case with divergence-free drift velocity $u \in L^{2}(\Omega)$ only depending on the spatial variable x. Finally, exploiting the Schauder fixed point theorem combined with the Arzelà-Ascoli compactness theorem, we obtain the existence and uniqueness of weak solutions in the standard Banach space $C([0,T]; H_{0}^{1}(\Omega))$ for a class of fractional superdiffusion equations.