Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),; u(0)=u_0\in \exp L^p(\R^N),$where$p>1$and$f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L_0^p({\mathbb{R}}^N)$ for $f(u)\sim \mbox{e}^{|u|^q},;0<q\leq p,; |u|\to \infty.$However,ifforsome$\lambda>0,$ $\displaystyle\liminf_{s\to \infty}\left(f(s),{\rm{e}}^{-\lambda s^p}\right)>0,$ then non-existence occurs in $\exp L^p({\mathbb{R}}^N)\mathrm{.}$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $\mathrm{\mid }f(u)\mathrm{\mid }\sim \mathrm{\mid }u{\mathrm{\mid }}^m$ as $u\to 0,$ $\frac{N(m-1)}{2}\ge p$, we show that the solution is global. In particular, $p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m$.