  Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the
problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - \Delta u \pm g(u)
&= \mu \quad &\text{in } Q_T:=\Omega \times (0,T) \\ \phantom{------,} u&=0
&\text{on } \partial \Omega \times (0,T)\\ \phantom{----,} u(.,0) &=\omega
&\text{in } \Omega. \end{array} \right. \end{equation} where $\mu$ and $\omega$
are bounded Radon measures in $Q_T$ and $\Omega$ respectively and $g(u) \sim
e^{a |u|^q} $ with $a>0$ and $q \geq 1$. We provide a sufficient condition in
terms of fractional maximal potentials of $\mu$ and $\omega$ for solving (P).
