Removable lateral singularities of semilinear parabolic PDEs and Besov capacities (Q1270414)

A closed set \(\Gamma\subset \mathbb{R}^d\) is called a removable singularity for the equation \(Lu= u^\alpha\), \(\alpha> 1\) if there is no non-trivial solution of this equation in \(\mathbb{R}^d\setminus\Gamma\). Suppose \(1<\alpha\leq 2\), that \(L\) is a second-order elliptic differential operator in \(\mathbb{R}^d\) and \(D\) is a bounded smooth domain in \(\mathbb{R}^d\). Let \(\Omega= \mathbb{R}_+\times D\), and let \(\Gamma\) be a compact set on the lateral boundary of \(\Omega\). The author proves that \(\Gamma\) is a removable lateral singularity for the equation \(\dot u+ Lu= u^\alpha\) in \(\Omega\) if and only if \(\text{cap}_{1/\alpha,2/\alpha,\alpha'}(\Gamma)= 0\) where cap stands for the Besov capacity on the boundary.