We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space $ L^r_{{\rm uloc}, \rho}( \Omega) $, where the solution is not decaying at $ |x|\to \infty $. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local $ L^r $ spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [
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