# On Agemi-type structural conditions for a system of semilinear wave equations

## TL;DR

This paper investigates a specific class of two-component cubic semilinear wave equations in two dimensions, demonstrating that solutions with small initial data become free waves as time progresses to infinity.

## Contribution

It establishes asymptotic freedom for solutions under Agemi-type conditions that do not satisfy previous structural assumptions, expanding understanding of wave behavior.

## Key findings

- Small amplitude solutions are asymptotically free as t→+∞
- The structural condition (Ag) ensures asymptotic freedom despite violating (Ag₀) and (Ag₊)
- Results apply to a class of two-component cubic semilinear wave systems

## Abstract

We consider a two-component system of cubic semilinear wave equations in two space dimensions satisfying the Agemi-type structural condition (Ag) but violating (Ag$_0$) and (Ag$_+$). For this system, we show that small amplitude solutions are asymptotically free as $t\to +\infty$.