Self-similar solutions of the porous medium equation in a half-space with a nonlinear boundary condition: existence and symmetry (Q1877811)

In this work, it is proved that the problem \[ \begin{cases} \Delta u = u^{\alpha} &\text{in}\quad \mathbb R^N_+\\ \frac{\partial u}{\partial\nu} = u &\text{on}\quad \partial\mathbb R^N_+\end{cases} \] where \(\partial/\partial\nu\) is the outer unit normal derivative and \(0<\alpha<1\), has a nonnegative solution with compact support. It is proved for this problem that the support of every nonnegative solution with a finite energy is compact and radially symmetric in tangential variables.