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We consider the wave equation with a cubic convolution ∂ 2 t u − ∆u = (|x| −γ * u 2 )u in three space dimensions. Here, 0 < γ < 3 and * stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then γ ≥ 2 assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when 2 ≤ γ < 3. In this paper, wedoi:10.3934/dcds.2021048 fatcat:lbl5ecknqvdr3dgpt3sukg4wo4