On the long time behaviour of the conical Kahler-Ricci flows

We prove that the conical Kahler-Ricci flows introduced in [11] exist for all time t is an element of [0, +infinity). These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C-1,C-beta is negative or zero, the corresponding conical Kahler-Ricci flows converge to Kahler-Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouvillc-type theorem for Kahler-Ricci flat metrics (which are defined over C-n) with conical singularities.