  We use the formalism of Hairer's regularity structures theory \cite{hai-14}
to study a heat equation with non-linear perturbation driven by a space-time
fractional noise. Different regimes are observed, depending on the global
pathwise roughness of the noise. To this end, and following the procedure
exhibited in \cite{hai-14}, the equation is first "lifted" into some abstract
"regularity structure" and therein solved through a fixed-point argument. Then
we construct a consistent "model" above the fractional noise, by relying on a
smooth Fourier-type approximation of the process.
