Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques

In this paper we are interested in the upper bound of the lifespan estimate for the compressible Euler system with time dependent damping and small initial perturbations. We employ some techniques from the blow-up study of nonlinear wave equations. The novelty consists in the introduction of tools from the Orlicz spaces theory to handle the nonlinear term emerging from the pressure $p \equiv p(\rho)$, which admits different asymptotic behavior for large and small values of $\rho-1$, being $\rho$ the density. Hence we can establish, in high dimensions $n\in\{2,3\}$, unified upper bounds of the lifespan estimate depending only on the dimension $n$ and on the damping strength, and independent of the adiabatic index $\gamma>1$. We conjecture our results to be optimal. The method employed here not only improves the known upper bounds of the lifespan for $n\in\{2,3\}$, but has potential application in the study of related problems.