We consider the Fujita equation $u_t=\Delta u+u^{p}$ on $\mathbb R^N$ with $N\ge 3$. We prove the existence of global, unbounded solutions for  $p_S<p<p_{JL}$, where $p_S:=(N+2)/(N-2)$, $p_{JL}=\infty$ if   $N\le 10$, and $p_{JL}:=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}$ if $N>10$. Previously, it was known that global, unbounded  solutions exist for $p\ge p_{JL}$, whereas for $p<p_S$ there are no such solutions.