Uniqueness of very weak solutions for a fractional filtration equation

We prove existence and uniqueness of distributional, bounded solutions to a fractional filtration equation in ${mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in cite{TJJ} that if two bounded solutions $u,w$ of eqref{eq6} satisfy $u-win L^1({mathbb R}^d imes(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions. For nonnegative initial data, we first show that a minimal solution exists and then that any other solution must coincide with it. A similar procedure is carried out for sign-changing solutions. As a consequence, distributional solutions have locally-finite energy.