  In this article, we consider the stochastic wave and heat equations on
$\mathbb{R}$ 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 H\"older 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 $p \geq 2$.
