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

## TL;DR

This paper proves that for certain wave equations outside the energy-critical case, the Sobolev norm of non-scattering solutions becomes unbounded at the maximal existence time, refining previous understanding.

## Contribution

It provides a unified proof that the scale-invariant Sobolev norm blows up for non-scattering solutions in energy-subcritical and supercritical wave equations, using the channel of energy method.

## Key findings

- Sobolev norm diverges at maximal time for non-scattering solutions
- Unified approach applicable to energy-subcritical and supercritical cases
- Introduction of weighted scale-invariant Sobolev spaces and profile decomposition

## Abstract

This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These spaces are local, thus adapted to finite speed of propagation, and related to a conservation law of the linear wave equation. We also construct the adapted profile decomposition.