  We consider a general form of a parabolic equation that generalizes both the
standard parabolic $p$-Laplace equation and the normalized version that has
been proposed in stochastic game theory. We establish an equivalence between
this equation and the standard $p$-parabolic equation posed in a fictitious
space dimension, valid for radially symmetric solutions. This allows us to find
suitable explicit solutions for example of Barenblatt type, and as a
consequence we settle the exact asymptotic behaviour of the Cauchy problem even
for nonradial data. We also establish the asymptotic behaviour in a bounded
domain. Moreover, we use the explicit solutions to establish the parabolic
Harnack's inequality.

Analysis of PDEs