Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times

TL;DR

This paper establishes lifespan estimates for solutions to Strauss type wave systems on asymptotically flat space-times, extending known results and allowing for non-compact initial data when exponents exceed 2.

Contribution

It introduces new space-time estimates related to local energy norms and applies them to derive sharp lifespan bounds for nonlinear wave systems.

Findings

Lower bounds for solution lifespan when p,q ≥ 2

Extension of global existence region for (p,q)

Sharp bounds in the subcritical case

Abstract

By assuming certain local energy estimates on $(1+3)$-dimensional asymptotically flat space-time, we study the existence portion of the \emph{Strauss} type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in \cite{MR2944027}. These estimates can be used to prove a series of weighted \emph{Strichartz} and \emph{KSS} type estimates, for wave equations on asymptotically flat space-time. Then we apply the space-time estimates to obtain the lower bound of the lifespan when the nonlinear exponents $p$ and $q\ge 2$. In particular, our bound for the subcritical case is sharp in general and we extend the known region of $(p,q)$ to admit global solutions. In addition, the initial data are not required to be compactly supported, when $p, q>2$.