Boundary regularity for fractional porous medium-type equations in bounded domains

We prove boundary Hölder regularity for fractional porous medium-type equations in bounded smooth domains with zero exterior data. For $0<\sigma <1$, $n>2\sigma$, and $m>1$, we show that nonnegative weak dual solutions $v$ of ${\mathrm{\partial }}_{t}v+(-\mathrm{\Delta }{)}^{\sigma }(v^m)=0$ satisfy $$\frac{v}{d^{\sigma \mathrm{/}m}}\in C_{x,t}^{\alpha ,\frac{\alpha m}{\sigma (m+1)}}$$ up to the boundary and away from the initial time, for some $\alpha \in (0,\sigma )$, where $d$ denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for $u\mathrm{/}{d}^{\sigma }$ for nonnegative viscosity solutions of ${\mathrm{\partial }}_{t}u=u^{\gamma }Lu$, with $0<\gamma <1$ and initial data comparable to $d^{\sigma }$. Here $L$ is a symmetric stable operator of order $2\sigma$ whose measurable angular density is bounded above and below by positive constants. Our proof combines comparison principles and boundary barriers with a blow-up and compactness argument based on a Liouville theorem for ancient solutions in a half-space.