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

Abstract

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 $$D_t^\alpha u - div\left( {u - {{\left( { - \Delta } \right)}^{ - \sigma }}u} \right) = f,0 < \sigma < \frac{1}{2}.$$ This paper poses the problem over {t ∈ R+, x ∈ R n } 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 Holder continuity for such weak solutions.