# New examples of probabilistic well-posedness for nonlinear wave equations

## TL;DR

This paper establishes probabilistic global well-posedness for fractional nonlinear wave equations with certain nonlinearities, demonstrating super-critical ill-posedness and exploring randomization effects.

## Contribution

It introduces new probabilistic well-posedness results for fractional wave equations with exponential or polynomial nonlinearities, including super-critical ill-posedness constructions.

## Key findings

- Proved global well-posedness on Gibbs measure support
- Constructed ill-posedness examples in super-critical regimes
- Extended results to general randomizations

## Abstract

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the results are truly super-critical in the considered functional setting. We also present a result in the case of a general randomisation in the spirit of the work by N. Burq and the second author.