Semilinear parabolic equations with measure boundary data and isolated singularities (Q1604973)

The very interesting paper under review deals with the semilinear parabolic equation \[ u_t-\Delta u+g(u)=0\quad \text{ in} Q_T=\Omega\times (0,T),\tag \(*\) \] where \(\Omega\subset {\mathbb R}^N\) is a bounded domain with smooth (e.g., \(C^2\)) boundary and \(g\in C^1({\mathbb R})\) is a non-decreasing function. Two main subjects are studied: The initial-boundary value problem for \((*)\) where the initial/boundary data is given by bounded (signed) measures. Precisely, setting \(\partial_\ell Q_T=\partial\Omega\times [0,T),\) \(\partial_0 Q_T=\Omega\times \{0\},\) the data is of the form \[ u=\nu\quad \text{ on} \partial_0 Q_T,\qquad u=\mu\quad \text{ on} \partial_\ell Q_T \] where \(\mu\) belongs to the space of bounded Borel measures on \(\partial_\ell Q_T,\) while \(\nu\) lies in the space of Radon measures on \(\Omega\) bounded with respect to the weight function \(\rho(x)=\text{ dist }(x,\partial\Omega).\) The second group of results regards classification and asymptotic behaviour of positive solutions to \((*)\) having an isolated point singularity on the lateral boundary.