Non-axial self-similar hole filling for the porous medium equation (Q2723514)

The article is devoted to studying the so-called focusing or hole-filling problem for the porous medium equation NEWLINE\[NEWLINE \partial_tu =\Delta(u^m),\quad m > 1. NEWLINE\]NEWLINE In the focusing problem the initial-value problem is solved for the above mentioned equation with data at \(t = 0\) whose support lies outside a compact set \(K\). The authors are concerned with the existence of non-axial self-similar solutions which bifurcate from the axially symmetric solutions. The aim of the article under review is to give a proof of the existence of these bifurcations. More specifically, the authors prove that as \(m\searrow 1\) the axially symmetric normalized self-similar solutions undergo an infinite sequence of symmetry breaking bifurcations. In particular, there exists infinitely many families of non-axial focusing self-similar solutions to the porous medium equation written in the terms of the scaled pressure function.