Is the stochastic parabolicity condition dependent on p and q ? ∗

In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus T = [0, 2π]. The equation is considered in L((0, T )×Ω;L(T)) for p, q ∈ (1,∞). 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.