Porous medium flow with both a fractional potential pressure and fractional time derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional and is of Caputo-type, which takes into account “memory”. The precise model is Dtαu−div(u−(−Δ)−σu)=f,0<σ<12.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_t^\alpha u - div\left( {u - {{\left( { - \Delta } \right)}^{ - \sigma }}u} \right) = f,0 < \sigma < \frac{1}{2}.$$\end{document} This paper poses the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x) have exponential decay at infinity is proved. The main result is Hölder continuity for such weak solutions.