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

## TL;DR

This paper constructs finite energy solutions to the focusing energy subcritical nonlinear wave equation that blow up precisely on any given compact set, using an ansatz based on ODE techniques and energy methods.

## Contribution

It introduces a novel method to produce solutions with blow-up exactly on arbitrary compact sets for the wave equation, expanding understanding of singularity formation.

## Key findings

- Solutions blow up exactly on specified compact sets.
- The construction relies on an ansatz involving ODE solutions.
- Energy and compactness methods confirm the solutions' properties.

## Abstract

We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - \Delta u= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. 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 $A\ge 0$ 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.