Abstract
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 \(m.\)