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

   
@article{EJP2186,
	author = {Zdzislaw Brzezniak and Mark Veraar},
	title = {Is the stochastic parabolicity condition dependent on $p$ and $q$?},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {stochastic parabolicity condition; parabolic stochastic evolution; multiplicative noise; gradient noise; blow-up; strong solution; mild solution; maximal regularity; stochastic partial differential equation},
	abstract = {
In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $\mathbb{T} = [0,2\pi]$. The equation is considered in $L^p((0,T)\times\Omega;L^q(\mathbb{T}))$ 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.
},
	pages = {no. 56, 1-24},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-2186},    
        url = {http://ejp.ejpecp.org/article/view/2186}}