Scattering for radial, bounded solutions of focusing supercritical wave equations (Q2878685)

The authors consider the wave equation in space dimension 3 with an energy supercritical, focusing nonlinearity, that is \(u_{tt}-\Delta u-|u|^{p-1} u=0\) with \(p>5\) and \(x\in\mathbb{R}^3\). The reason that why it is called energy supercritical and focusing is simply from the fact that it admits a formal conservation of energy \(\int_{\mathbb{R}^3} |u_t|^2+|\nabla u|^2-\frac{2}{p+1} |u|^{p+1} dx=c\) (noticing the minus sign) and we do not have the Sobolev embedding of the form \(H^1\subset L^{p+1}\). In addition, the problem is critical according to the space \(\dot H^{s_p}\) with \(s_p=3/2-2/(p-1)\). They show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and scatters to a linear solution. As a consequence, finite time blow-up solutions have critical Sobolev norm \(\dot H^{s_p}\) converging to infinity (along some sequence of times). This is in stark contrast to the energy critical case, where radial type II blow-up solutions exist. The proof relies on the compactness/rigidity method, pointwise estimates on compact solutions obtained by the last two authors, and channels of energy arguments used by the authors in previous works on the energy-critical equation.