Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $du=(a^{ij}{u}_{x^i{x}^j}+b^i{u}_{x^i}+cu)dt+\xi {\left|u\right|}^{1+\lambda }dF,(t,x)\in (0,\infty )\times {\mathbb{R}}^d,$ where $\lambda \ge 0$ and the coefficients depend on $(\omega ,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $\lambda$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$.

对于带有彩色噪声的随机PDE，可以获得强解的存在性，唯一性和规则性 $F$ 及其超线性扩散系数： $dü=（{\mathrm{一种}}^{\mathrm{一世}Ĵ}{ü}_{X^{\mathrm{一世}}{X}^{Ĵ}}+b^{\mathrm{一世}}{ü}_{X^{\mathrm{一世}}}+Cü）dŤ+\xi {\left|ü\right|}^{\mathrm{1个}+\lambda }dF，（Ť，X）\in （0，\infty ）\times {\mathbb{[R}}^d，$ 在哪里 $\lambda \ge 0$ 系数取决于 $（\omega ，Ť，X）$。处理扩散系数非线性的策略是为一般的Lipschitz情况找到一个清晰的估计，并将其应用于超线性情况。此外，对估算的调查提供了$\lambda$，这是获得独特可溶性的充分条件，其中范围取决于...的空间协方差 $F$ 和空间维度 $d$。