Local regularity and finite time singularity for the generalized SQG equation on the half-plane

We show that the generalized SQG equation with $\alpha \in (0,\frac{1}{4}]$ is locally well-posed on the half-plane in spaces of bounded integrable solutions that are natural for its dynamic on domains with boundaries, and allow for some power growth of the solution derivative in the normal direction at the boundary. We also show existence of solutions exhibiting finite time blow-up in the whole local well-posedness parameter regime $\alpha \in (0,\frac{1}{4}]$, which is the first finite time singularity result for equations (as opposed to patch models) of this type. Moreover, we prove optimality of both these results by showing ill-posedness of the PDE in all the above spaces when $\alpha >\frac{1}{4}$.