Local existence for evolution equations with nonlocal term in time and singular initial data

We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))dsin Ω × (0 , T) , where 0 < α≤ 2 , m is a nonnegative and measurable homogeneous function defined on K= { (t, s) ∈ R², 0 < s< t} , f is a nonnegative, continuous and nondecreasing function and Ω is either a bounded smooth domain or the whole space R^N. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space L^r(Ω) , 1 ≤ r< ∞.