Is the stochastic parabolicity condition dependent on p and q?

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In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus =[0,2π]. The equation is considered in Lᵖ(Ø×(0,T);Lq()) 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.