  Nonlinear heat equations in two dimensions with singular initial data are
studied. In recent works nonlinearities with exponential growth of
Trudinger-Moser type have been shown to manifest critical behavior:
well-posedness in the subcritical case and non-existence for certain
supercritical data. In this article we propose a specific model nonlinearity
with Trudinger-Moser growth for which we obtain surprisingly complete results:
a) for initial data strictly below a certain singular threshold function
$\widetilde u$ the problem is well-posed, b) for initial data above this
threshold function $\widetilde u$, there exists no solution, c) for the
singular initial datum $\widetilde u$ there is non-uniqueness. The function
$\widetilde u$ is a weak stationary singular solution of the problem, and we
show that there exists also a regularizing classical solution with the same
initial datum $\widetilde u$.
