The Cauchy problem for the parabolic equation $u_t=Delta u+u^{1+alpha} (alpha>0)$ is considered. It is well known that, the given equation being nonlinear, the solution need not exist for all time $t$. The influence of the dimension $m$ of the space variable $x$ on the existence or nonexistence of the solution for all time $t$ is investigated. Particularly interesting is the result according to which, for $0