Existence and uniqueness results for a time-fractional nonlinear diffusion equation

In this work we consider a nonlinear ordinary integro-differential equation which arises in the studies of time-fractional porous medium equation. The nonlocality of the resulting free-boundary problem is governed by the Erdélyi-Kober operator which requires using other than classical proof techniques. To prove the existence and uniqueness of a compactly supported solution we reduce the free-boundary case to the initial-value problem. Next, we use the sub- and supersolution technique to show that there exists a globally defined unique solution. As a side product, some estimates on the exact solution are found.