Is the stochastic parabolicity condition dependent on \(p\) and \(q\)? (Q456200)

The well-posedness of the SPDE NEWLINE\[NEWLINE\begin{aligned} du(t)& = \Delta u(t,x)dt+2\alpha Du(t,x)dW(t)+2\beta|D|u(t,x)dW(t),\\ u(0,x)&=u_0(x)\end{aligned}\tag{1} NEWLINE\]NEWLINE on the torus \(\mathbb{T}=[0,2\pi]\) and for \(t\in\mathbb{R}_+\), with \(D=(-\Delta)^{1/2}\) is studied in the spaces \(L^{p}(\Omega,L^{q}(\mathbb{T}))\) for \(p,q\in(1,\infty).\) It is shown that (1) is well-posed in \(L^{p}(\Omega,L^{q}(\mathbb{T}))\) if NEWLINE\[NEWLINE 2\alpha^{2}+2\beta^{2}(p-1)<1 NEWLINE\]NEWLINE and ill-posed ifNEWLINENEWLINENEWLINE\[NEWLINE 2\alpha^{2}+2\beta^{2}(p-1)>1. NEWLINE\]NEWLINE The results are compared to the ``classical'' \(p\)-independent stochastic parabolicity condition NEWLINE\[NEWLINE 2\alpha^{2}+2\beta^{2}<1. NEWLINE\]NEWLINE In fact, the well-posedness result is given for more general, abstract SPDE of the form NEWLINE\[NEWLINE\begin{aligned} dU(t)+AU(t)dt &= 2BU(t)dW(t),\quad t\in\mathbb{R}_+,\\ U(0)& = u_{0},\end{aligned}NEWLINE\]NEWLINE with \(A=CC^{*}\), \(B=\alpha C+\beta|C|\) for some \(\alpha,\beta\in\mathbb{R}\) and some skew-adjoint operator \(C.\)