Abstract
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 in the case of
monomial nonlinearity in dimension $d\geq3$. Next, 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.