Regularity and Capacity for the Fractional Dissipative Operator
  This note is devoted to exploring some analytic-geometric properties of the regularity and capacity associated to the so-called fractional dissipative operator $\partial_t+(-\Delta)^\alpha$, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation $(\partial_t+(-\Delta)^\alpha)u(t,x)=F(t,x)$ subject to $u(0,x)=0$. 
Renjin Jiang, Jie Xiao, Dachun Yang, and Zhichun Zhai
math.AP math.FA