A time-space fractional parabolic type problem: weak, strong and classical solutions

Abstract

We use a generalized Riemann-Liouville type derivative of an abstract function of one variable and existence of a weak solution to an abstract fractional parabolic problem on [0, T] containing Riemann-Liouville derivative of a function of one variable and spectral fractional powers of a weak Dirichlet-Laplace operator to study existence of a strong solution to this problem. Our goal in this regard is to provide conditions that allow the transition from a weak to a strong solution. Next, we passage from the abstract problem to a classical one on \([0,T]\times \varOmega \), containing partial (with respect to time \(t\in [0,T]\,\)) Riemann-Liouville derivative of the unknown real-valued function of two variables and fractional powers of a weak Dirichlet-Laplacian of this function (with respect to spatial variable \(x\in \varOmega \)). The most important in this regard is a theorem on the relation of the fractional derivatives of an abstract function of one variable and real-valued one of two variables.