Global existence for a Fritz John equation in expanding FLRW spacetimes

We study the family of semilinear wave equations $\square_{\mathbf{g}_p}\phi=(\partial_t\phi)^2$, on fixed expanding FLRW spacetimes, having $\mathbb{R}^3$ spatial slices and undergoing a power law expansion, with scale factor $a(t)=t^p$, $0< p \le 1$. This is a natural generalization to a non-stationary background of a famous Fritz John ''blow-up'' equation in $\mathbb{R}^{1+3}$ (corresponding to $p=0$, i.e. the case in which $\mathbf{g}_0$ is the Minkowski metric). While, in Minkowski spacetime ($p=0$), non-trivial solutions to this equation are known to diverge in finite time, here we prove that, on the referred FLRW backgrounds ($0<p\leq 1$), sufficiently small, smooth, and compactly supported initial data yield global-in-time solutions to the future. Previous work, co-authored by the first two authors, considered accelerated expanding spacetimes ($p>1$) and relied on the integrability of the inverse of the scale factor to establish future global well-posedness. In the current work, where such an integrability condition is lacking, we rely on a vector field method that captures and combines dispersive estimates with the spacetime expansion to control the solution and suppress the nonlinear blow-up mechanism. To achieve this, we commute the Laplace-Beltrami operator with a boosts-free subset of the Poincar\'e algebra and employ Klainerman-Sideris types of inequalities. Our strategy is general and is developed to handle the non-stationary nature of FLRW spacetimes. While we focus solely on this Fritz John type of equation, which serves as a prototype to study blow-up of non-linear waves, our approach provides a rigorous proof of the regularizing effects of spacetime expansion and can be exploited for a wider range of applications and nonlinearities.