A note on nonexistence of global solutions to a nonlinear integral equation M. Guedda M. Kirane Abstract In this paper we study the Cauchy problem for the integral equation u t = −(−Δ) β 2 u + h(t)u 1+α in R N × (0,T ), where 0 <β ≤ 2. We obtain some extension of results of Fujita who considered the case β = 2 and h ≡ 1. 1 Introduction This article deals with the blow-up of positive solutions to the Cauchy problem for the integrodifferential equation u t = −(−Δ) β 2 u + h(t)u 1+α in R N × (0,T ), (1.1) u(x, 0) = u 0 (x) ≥ 0 for x ∈ R N , (1.2) where (−Δ) β 2 , for 0 <β ≤ 2, denote the fractional power of the operator −Δ. It is assumed that u 0 is a continuous function defined on R N and α is a positive constant. The function h satisfies h 1 ) h ∈ C [0, ∞),h ≥ 0, h 2 ) c 0 t σ ≤ h(t) ≤ c 1 t σ for sufficiently large t, where c 0 ,c 1 > 0 and σ> −1 are constants. Received by the editors February 1998. Communicated by J. Mawhin. 1991 Mathematics Subject Classification : 35K20, 35K55. Bull. Belg. Math. Soc. 6 (1999), 491–497