Sharp lifespan results for plane-symmetric fluids on decelerated spacetimes

In this article, we analyze plane-symmetric solutions to the relativistic Euler equations on spatially flat spacetimes with decelerated expansion. We consider a linear equation of state $p(\rho )=K\rho$ and power law inflation $a(t)=t^{\alpha }$, where $0<K<\frac{1}{3}$ and $0\le \alpha <1$. Previous results imply nonlinear stability of the homogeneous and isotropic solution above an expansion threshold ${\alpha }_{\text{crit}}=\frac{2}{3(1-K)}$. We prove that at and below this critical expansion rate there exist arbitrarily small data that exhibit finite-time singularity formation of the shock type, i.e., gradient blowup while the solution stays bounded. For this type of data, we establish lifespan estimates that are polynomial below and exponential at the critical threshold. In addition, we prove a theorem based on energy estimates that provides lower bounds on the lifespan of plane-symmetric solutions generated from small initial data and generalizes to solutions without symmetry in $3+1$-dimensions.