  In this paper we consider the initial value {problem $\partial_{t} u- \Delta
u=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f :
\mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with
$f(0)=0.$ Under smallness condition on the initial data and for nonlinearity
$f$ {such that $|f(u)|\sim \mbox{e}^{|u|^q}$ as $|u|\to \infty$,} $|f(u)|\sim
|u|^{m}$ as $u\to 0,$ $0<q\leq p\leq\,m,\;{N(m-1)\over 2}\geq p>1$, we show
that the solution is global. Moreover, we obtain decay estimates in Lebesgue
spaces for large time which depend on $m.$
