  In this paper we study well-posedness of a second order SPDE with
multiplicative noise on the torus $\T =[0,2\pi]$. The equation is considered in
$L^p(\O\times(0,T);L^q(\T))$ for $p,q\in (1, \infty)$. It is well-known that if
the noise is of gradient type, one needs a stochastic parabolicity condition on
the coefficients for well-posedness with $p=q=2$. In this paper we investigate
whether the well-posedness depends on $p$ and $q$. It turns out that this
condition does depend on $p$, but not on $q$. Moreover, we show that if $1<p<2$
the classical stochastic parabolicity condition can be weakened.
