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

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]x Omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,T]\times \varOmega $$\end{document}, containing partial (with respect to time t is an element of[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T]\,$$\end{document}) 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 is an element of Omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \varOmega $$\end{document}). 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.