We consider the parabolic system ut−aΔu=f(v),vt−bΔv=g(u) in Ω×(0,T), where a,b>0, f,g:[0,∞)→[0,∞) are non-decreasing continuous functions and either Ω is a bounded domain with smooth boundary ∂Ω or the whole space RN. We characterize the functions f and g so that the system has a local solution for every initial data (u0,v0)∈Lr(Ω)×Ls(Ω), u0,v0≥0, r,s∈[1,∞).