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 $D_t^α u - \text{div}(u(-Δ)^{-σ} u) = f, \quad 0<σ<1/2.$ We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0$ as well as right hand side $f\geq 0$. 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.

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