Nontrivial compact blow-up sets of lower dimension in a half-space

Abstract

In this paper we provide examples of blowing-up solutions to parabolic problems in a half space, ${{\mathbb{R}}}^N_+ \times {{\mathbb{R}}}^M = \{x_N >0 \} \times {{\mathbb{R}}}^M$, with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we prove existence of a nontrivial compactly supported solution to $\nabla (|\nabla \varphi|^{p-2} \nabla \varphi) = \varphi $ in the half space ${{\mathbb{R}}}^N_+ =\{x_N >0\}$ with the nonlinear boundary condition $-|\nabla \varphi|^{p-2} \frac{\partial \varphi}{\partial x_N} = \varphi^{p-1}$ on $\partial {{\mathbb{R}}}^N_+ =\{ x_N =0\}$.