SPDEs with affine multiplicative fractional noise in space with index 1/4 < H < 1/2

In this article, we consider the stochastic wave and heat equations on R with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index H, with 1/4 < H < 1/2. We assume that the diffusion coefficient is given by an affine function sigma(x) = ax + b, and the initial value functions are bounded and Holder continuous of order H. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is L-2 (Omega) -continuous and its p-th moments are uniformly bounded, for any p >= 2