Global existence and decay estimates for the heat equation with exponential nonlinearity

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 $u\to 0,$ $0<q\le p\le m,\frac{N(m-1)}{2}\ge p>1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$

To appear. arXiv admin note: text overlap with arXiv:1607.02723, arXiv:1606.07320