Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as well as right hand side f ≥ 0. We first prove existence for weak solutions when f, u(0, x) have exponential decay at infinity. Our main result is Hölder continuity for such weak solutions.