Narrow Results

In this paper we consider the problem: ∂_tu − Δu = f (u), u(0) = u₀ ∈ exp L^p (ℝ^N), where p > 1 and f : ℝ → ℝ having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in $\exp L_0^p\left({ℝ}^N\right)$ for $f\left(u\right)\sim e^{{\left|u\right|}^q},0<q\le p,\left|u\right|\to \infty$. However, if for some λ > 0, $\underset{s\to \infty }{\lim }\inf \left(f\left(s\right)e^{-\lambda s^p}\right)>0$ 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)| ∼ |u|^m as u → 0, $\frac{N\left(m-1\right)}{2}\ge p$, 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.