# Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

## TL;DR

This paper investigates the conditions under which solutions to a weakly coupled system of damped semilinear wave equations blow up, using advanced iteration and slicing methods to analyze subcritical and critical cases with power nonlinearities.

## Contribution

It introduces a novel approach combining lower bounds, slicing methods, and auxiliary functions to determine blow-up conditions in the critical case.

## Key findings

- Identifies the critical curve for exponents (p, q) matching that of undamped systems.
- Develops an iteration method for subcritical cases based on space average bounds.
- Adapts the slicing method with weight functions for the critical case analysis.

## Abstract

In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p, q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities.