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

Abstract Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise F and its super-linear diffusion coefficient: d u = ( a i j u x i x j + b i u x i + c u ) d t + ξ | u | 1 + λ d F , ( t , x ) ∈ ( 0 , ∞ ) × R d , where λ ≥ 0 and the coefficients depend on ( ω , 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 λ , a sufficient condition for the unique solvability, where the range depends on the spatial covariance of F and the spatial dimension d .