Regularity and capacity for the fractional dissipative operator

This paper is devoted to exploring some analytic geometric properties of the regularity and capacity associated with the so-called fractional dissipative operator partial derivative(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 of (partial derivative(t)+ (-Delta)(alpha))u(t,x) = F(t, x) subject to u(0, x) = 0. The methods used in this paper rely on effectively controlling the time-dependent non-local kernels and potentials with fractional order alpha is an element of (0, 1), dual representation of the capacity and Frostman type theorem from geometric measure theory. (C) 2015 Elsevier Inc. All rights reserved.