  We consider a family of nonlinear stochastic heat equations of the form
$\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes
space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy
process on $\R$, and $\sigma$ is Lipschitz continuous and zero at 0. We show
that this stochastic PDE has a random-field solution for every finite initial
measure $u_0$. Tight a priori bounds on the moments of the solution are also
obtained.
  In the particular case that $\mathcal{L}f=cf"$ for some $c>0$, we prove that
if $u_0$ is a finite measure of compact support, then the solution is with
probability one a bounded function for all times $t>0$.
