In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $mathbb{T} = [0,2pi]$. The equation is considered in $L^p((0,T)imesOmega;L^q(mathbb{T}))$ for $p,qin (1, infty)$. 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 $1<2$ the classical stochastic parabolicity condition can be weakened.
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机译：在本文中，我们研究了在圆环$ mathbb {T} = [0,2 pi] $上具有乘性噪声的二阶SPDE的适定性。对于$ p，q in（1， infty）$，方程在$ L ^ p（（0，T） times Omega; L ^ q（ mathbb {T}））$中考虑。众所周知，如果噪声是梯度型的，则需要在具有适定性的系数上具有$ p = q = 2 $的随机抛物线条件。在本文中，我们研究了适定性是否取决于$ p $和$ q $。事实证明，这种情况确实取决于$ p $，而不取决于$ q $。此外，我们表明如果$ 1 <2 $，则经典随机抛物线条件可以被削弱。
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