  We consider a system of d non-linear stochastic heat equations in spatial
dimension 1 driven by d-dimensional space-time white noise. The non-linearities
appear both as additive drift terms and as multipliers of the noise. Using
techniques of Malliavin calculus, we establish upper and lower bounds on the
one-point density of the solution u(t,x), and upper bounds of Gaussian-type on
the two-point density of (u(s,y),u(t,x)). In particular, this estimate
quantifies how this density degenerates as (s,y) converges to (t,x). From these
results, we deduce upper and lower bounds on hitting probabilities of the
process {u(t,x)}_{t \in \mathbb{R}_+, x \in [0,1]}, in terms of respectively
Hausdorff measure and Newtonian capacity. These estimates make it possible to
show that points are polar when d >6 and are not polar when d<6. We also show
that the Hausdorff dimension of the range of the process is 6 when d>6, and
give analogous results for the processes t \mapsto u(t,x) and x \mapsto u(t,x).
Finally, we obtain the values of the Hausdorff dimensions of the level sets of
these processes.
