Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations

Abstract

We consider a wave equation in three space dimensions, with a power-like nonlinearity which is either focusing or defocusing. The exponent is greater than 3 (conformally supercritical) and not equal to 5 (not energy-critical). We prove that for any radial solution which does not scatter to a linear solution, an adapted scale-invariant Sobolev norm goes to infinity at the maximal time of existence. The proof uses a conserved generalized energy for the radial linear wave equation, new Strichartz estimates adapted to this generalized energy, and a bound from below of the generalized energy of any nonzero solution outside wave cones. It relies heavily on the fact that the equation does not have any nontrivial stationary solution. Our work yields a qualitative improvement on previous results on energy-subcritical and energy-supercritical wave equations, with a unified proof.