Strong existence and uniqueness to a class of nonlinear SPDEs driven by Gaussian colored noises

In this paper, we establish the strong existence and uniqueness to the following stochastic partial differential equation: ut(x)=u0(x)+12∫0tΔus(x)ds+∫0t∫Rg(us(x),∇us(x))dF(s,y),where F(s,x) is a Gaussian noise on R+×R that is white in time and colored in space. This gives an affirmative answer to the open problem in Gomez et al. (2013) for certain non-Lipschitz condition of g.