  We investigate the initial value problem for a semilinear heat equation with
exponential-growth nonlinearity in two space dimension. First, we prove the
local existence and unconditional uniqueness of solutions in the Sobolev space
$H^1(\R^2)$. The uniqueness part is non trivial although it follows
Brezis-Cazenave's proof \cite{Br} in the case of monomial nonlinearity in
dimension $d\geq3$. Next, %Following Caffarelli-Vasseur \cite{cv}, we show that
in the defocusing case our solution is bounded, and therefore exists for all
time. In the focusing case, we prove that any solution with negative energy
blows up in finite time. Lastly, we show that the unconditional result is lost
once we slightly enlarge the Sobolev space $H^1(\R^2)$. The proof consists in
constructing a singular stationnary solution that will gain some regularity
when it serves as initial data in the heat equation. The Orlicz space appears
to be appropriate for this result since, in this case, the potential term is
only an integrable function.
