Self-similar solutions for a fractional thin film equation governing hydraulic fractures

In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient $K\geq 0$. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness ($K=0$) and the finite toughness $K\in(0,\infty)$ cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all $K\textgreater{}0$ there exists an injection rate for the fluid such that self-similar solutions exist.