On the Cauchy problem for the generalized porous medium equation

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) = Delta phi (u) under broad assumptions on phi. In particular, functions phi (s) similar to s(m) lnP s, m greater than or equal to 1, p greater than or equal to -1, and phi (s) similar to exp(s(p)), p > 0, (as s --> +infinity) are included. We give sufficient conditions on the growth of the initial data as /x/ --> infinity, 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 cp we prove the uniqueness of a weak solution.