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 RN. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω) , 1 ≤ r< ∞.