A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $\lambda \geq 0$ 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 $d$.