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 $\exp L^p_0(\R^N)$ for $f(u)\sim \mbox{e}^{|u|^q},\;0<q\leq p,\; |u|\to \infty.$ However, if for some $\lambda>0,$ $\displaystyle\liminf_{s\to \infty}\left(f(s)\,{\rm{e}}^{-\lambda s^p}\right)>0,$ then non-existence occurs in $\exp L^p(\R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^{m}$ as $u\to 0,$ ${N(m-1)\over 2}\geq p$, 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 $m$.