  We consider the white-noise driven stochastic heat equation on
$[0,\infty)\times[0,1]$ with Lipschitz-continuous drift and diffusion
coefficients $b$ and $\sigma$. We derive an inequality for the
$L^1([0,1])$-norm of the difference between two solutions. Using some
martingale arguments, we show that this inequality provides some {\it a priori}
estimates on solutions. This allows us to prove the strong existence and
(partial) uniqueness of weak solutions when the initial condition belongs only
to $L^1([0,1])$, and the stability of the solution with respect to this initial
condition. We also obtain, under some conditions, some results concerning the
large time behavior of solutions: uniqueness of the possible invariant
distribution and asymptotic confluence of solutions.
