Generically sharp decay and blowing up at infinity for a weak null wave system

TL;DR

This paper establishes sharp pointwise decay estimates for solutions to a weak null wave system, revealing generic energy growth and frequency cascade phenomena akin to behaviors in Einstein's equations.

Contribution

It provides the first precise decay bounds for a weak null system, demonstrating sharpness and revealing energy blow-up and cascade effects for small data solutions.

Findings

Sharp decay estimates for solutions and their leading terms

Generic energy growth to infinity as time progresses

Evidence of energy cascade from high to low frequencies

Abstract

We study a system of semilinear wave equations satisfying the weak null condition, which can be regarded as a simplified model for the Einstein vacuum equations. The main objective is to establish precise pointwise decay estimates, as both lower and upper bounds of decay, for small data solutions. Specifically, we show that the difference between the solution and its leading-order term is dominated by lower-order terms that decay faster in the retarded time variable $u=t-r$. Moreover, we prove that these pointwise decay estimates are sharp for a generic class of small initial data decaying sufficiently fast. As applications of these estimates, we demonstrate that the energy of one component of the solution admits a lower bound that generically grows to infinity as $t\to +\infty$, which can be interpreted as ``blowing up at infinity." Furthermore, we verify that this component generically exhibits an energy cascade from high to low frequencies.