# A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms

## TL;DR

This paper proves blow-up results for a weakly coupled semilinear wave system with scale-invariant terms, using iteration and test function methods, and conjectures a shift in the critical curve for the exponents.

## Contribution

It introduces new blow-up results in both subcritical and critical cases and proposes a conjecture on the critical curve shift for the coupled system.

## Key findings

- Blow-up results established for subcritical and critical cases.
- Use of iteration argument and test function method in proofs.
- Conjecture on the shift of the critical curve for the exponents.

## Abstract

In this note two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case, when the damping and the mass terms make both equations in some sense "wave-like". In the proof of the subcritical case an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case we employ a test function type method, that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p - q plane of the exponents of the power nonlinearities for this weakly coupled system we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.