# Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping

## TL;DR

This paper investigates how a negative mass term causes finite-time blow-up in solutions to semilinear wave equations with scattering damping, especially when the coefficient decay is slow, leading to shorter lifespan estimates.

## Contribution

It demonstrates the dominant effect of the negative mass term on blow-up and provides new, shorter lifespan estimates for solutions with small initial data.

## Key findings

- Negative mass term induces finite-time blow-up.
- Lifespan estimates are significantly shorter than classical results.
- Blow-up occurs when the decay of the mass term's coefficient is slow.

## Abstract

In this paper we study blow-up and lifespan estimate for solutions to the Cauchy problem with small data for semilinear wave equations with scattering damping and negative mass term. We show that the negative mass term will play a dominant role when the decay of its coefficients is not so fast, thus the solutions will blow up in a finite time. What is more, we establish a lifespan estimate from above which is much shorter than the usual one.