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In this article, we consider the stochastic wave and heat equations on$\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noisewhich is white in time and behaves in space like a fractional Brownian motionof index $H$, with $1/4<H<1/2$. We assume that the diffusion coefficient isgiven by an affine function $\sigma(x)=ax+b$, and the initial value functionsare bounded and H\"older continuous of order $H$. We prove the existence anduniqueness of the mild solution for both equations. We show that the solutionis $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, forany $p \geq 2$.

SPDEs with fractional noise in space with index $H<1/2$