Existence of solutions for a fractional semilinear parabolic equation with singular initial data

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem ∂tu+(−Δ)θ2u=up,x∈RN,t>0,u(0)=μ≥0 in RN,where N≥1, 0<θ≤2, p>1 and μ is a Radon measure or a measurable function in RN. Our conditions lead optimal estimates of the life span of the solution with μ behaving like λ|x|−A (A>0) at the space infinity, as λ→+0.