Regularity and capacity for the fractional dissipative operator

Abstract

This paper is devoted to exploring some analytic–geometric properties of the regularity and capacity associated with the so-called fractional dissipative operator ∂t+(−Δ)α, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation (∂t+(−Δ)α)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 α∈(0,1), dual representation of the capacity and Frostman type theorem from geometric measure theory.