Type II singularities on complete non-compact Yamabe flow

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

Regularity Estimates for the p-Sobolev Flow

We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case $$p=2$$p=2. In this article we establish a priori estimates and regularity results for the p-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.