An $L_p$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

We obtain uniqueness and existence of a solution $u$ to the following second-order stochastic partial differential equation (SPDE) : \begin{align} \label{abs eqn} du= \left( \bar a^{ij}(\omega,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot)=0, \end{align} where $T\in (0,\mathrm{\infty })$, $w^k$ $(k=1,2,\dots )$ are independent Wiener processes, $({\bar{a}}^{ij}(\omega ,t))$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\kappa \mathrm{\mid }\xi {\mathrm{\mid }}^2\le {\bar{a}}^{ij}(\omega ,t){\xi }^i{\xi }^j\le K\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\forall }(\omega ,t,\xi )\in \mathrm{\Omega }\times (0,T)\times {\mathbf{R}}^d$$ for some $\kappa ,K\in (0,\mathrm{\infty })$, $$f\in L_p((0,T)\times {\mathbf{R}}^d,dt\times dx;L_r(\mathrm{\Omega },\mathcal{F},dP)),$$ and $$g,g_x\in L_p((0,T)\times {\mathbf{R}}^d,dt\times dx;L_r(\mathrm{\Omega },\mathcal{F},dP;l_2))$$ with $2\le r\le p<\mathrm{\infty }$ and appropriate measurable conditions.