We prove the existence of a weak solution of the Cauchy problem in classes of growing functions for the generalized porous medium equation u t = Δϕ(u) under broad assumptions on ϕ. In particular, functions ϕ(s) ∼ s m ln p s, m ≥ 1, p ≥ −1, and ϕ(s) ∼ exp(s p ), p > 0, (as s → +∞) are included. We give sufficient conditions on the growth of the initial data as |x| → ∞, which, in general, can not be improved, as we illustrate by examples. A lower bound on the existence time is also obtained. Under the convexity assumption on ϕ we prove the uniqueness of a weak solution.