  We study the nonlinear stochastic heat equation in the spatial domain
$\mathbb {R}$, driven by space-time white noise. A central special case is the
parabolic Anderson model. The initial condition is taken to be a measure on
$\mathbb {R}$, such as the Dirac delta function, but this measure may also have
noncompact support and even be nontempered (e.g., with exponentially growing
tails). Existence and uniqueness of a random field solution is proved without
appealing to Gronwall's lemma, by keeping tight control over moments in the
Picard iteration scheme. Upper bounds on all $p$th moments $(p\ge2)$ are
obtained as well as a lower bound on second moments. These bounds become
equalities for the parabolic Anderson model when $p=2$. We determine the growth
indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152
(2012) 681-701].
