  This paper concerns with the heat equation in the half-space
$\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary
$\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using
Kato and Hardy inequalities) that allows us to consider critical potentials
with infinite many singularities and anisotropy. Motivated by potential
profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which
we prove key linear estimates for some boundary operators arising from the
Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate
qualitative properties of solutions like self-similarity, positivity and
symmetry around the axis $\overrightarrow{Ox_{n}}$.
