# Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms

## TL;DR

This paper investigates the blow-up of solutions in a coupled system of semilinear damped wave equations with mixed nonlinearities, establishing critical conditions for blow-up and improving lifespan estimates using iteration and slicing methods.

## Contribution

It introduces a new approach combining iteration and slicing methods to analyze blow-up in a coupled damped wave system with mixed nonlinearities, extending previous results.

## Key findings

- Identifies the critical curve in the p-q plane for blow-up.
- Proves blow-up on the critical curve using a novel approach.
- Provides improved lifespan estimates in the non-damped case.

## Abstract

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the p-q plane for the pair of exponents (p,q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.