We consider the parabolic system u(t) - a Delta u = f (v), v(t) - b Delta v = g(u) in Omega x (0, T), where a, b > 0, f, g : [0, infinity) -> [0, infinity) are non-decreasing continuous functions and either Omega is a bounded domain with smooth boundary partial derivative Omega or the whole space R-N. We characterize the functions f and g so that the system has a local solution for every initial data (u(0), v(0)) is an element of L-r (Omega) x L-s (Omega), u(0), v(0) >= 0, r, s is an element of [1, infinity). (C) 2019 Elsevier Inc. All rights reserved.