We consider the semilinear equation ut+(-Δ)α/2u=∫0tm(t,s)f(u(s))ds\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u_t+(-\Delta )^{\alpha /2}u=\int \limits _0^t m(t,s) f(u(s)) \mathrm{d}s \end{aligned}$$\end{document}in Ω×(0,T)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \times (0,T)$$\end{document}, where 0<α≤2\documentclass[12pt]{minimal}
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\begin{document}$$0<\alpha \le 2$$\end{document}, m is a nonnegative and measurable homogeneous function defined on K={(t,s)∈R2,0<s<t}\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {K}}= \{ (t,s) \in {\mathbb {R}}^2, 0<s<t \}$$\end{document}, f is a nonnegative, continuous and nondecreasing function and Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is either a bounded smooth domain or the whole space RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}. Our goal is to determine conditions for the local existence and nonexistence of solutions with nonnegative initial data belonging to the space Lr(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$L^r(\Omega )$$\end{document}, 1≤r<∞\documentclass[12pt]{minimal}
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\begin{document}$$1 \le r < \infty $$\end{document}.
