Solutions blowing up on any given compact set for the energy subcritical wave equation

We consider the focusing energy subcritical nonlinear wave equation $partial_{tt} u - Delta u= |u|^{p-1} u$ in ${mathbb R}^N$, $Nge 1$. Given any compact set $ E subset {mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = kappa (t + A(x) )^{ -frac {2} {p-1} }$, where $Age 0$ vanishes exactly on $ E$, which is a solution of the ODE $h = h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen $A$), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.