Finite-time blow-up conjecture for the damped compressible Euler equations

Let $w$ solve the equation denoted by

, with smooth initial data $(w,\partial_t w)|_{t=0}$. The parameters satisfy

\lambda=1,\quad 0<\mu\leq 2,\qquad\text{or}\qquad \lambda>1,\quad \mu>0.

$$**Finite-time blow-up conjecture.** The smooth solution of$$

will blow up in finite time for a family of smooth initial data $(w,\partial_t w)|_{t=0}$, even if $(w,\partial_t w)|_{t=0}$ is sufficiently small.

This conjecture concerns the complementary parameter regimes to those in which the paper proves global existence and convergence to a modified Barenblatt solution. It is motivated by previously established finite-time blow-up results under small-perturbation assumptions, and the authors indicate that the conjecture will be considered in future work.