The Majda-Rosales-Schonbek (MRS) system, obtained through an asymptotic expansion of the non-isentropic compressible Euler equations in one space dimension, describes resonant interactions of small amplitude, spatially periodic sound waves with a small amplitude entropy wave. In particular, we are interested in when a small-amplitude smooth solution develops singularities. Previous work has shown that for small initial data or order $\epsilon$, the solution will remain smooth for time $O(\frac{\log \epsilon}{\epsilon})$. However, numerical simulations suggest the lifespan of small initial data can be extended to $O(\frac{1}{\epsilon^2})$. In this paper, we will explore multiple approaches that are often used to prove extended lifespans of quadratically quasilinear PDEs, which usually corresponds to an $O(\frac{1}{\epsilon})$ lifespan of small smooth solutions.

An asymptotic expansion of the MRS system produces a Degenerate Quasilinear Schrodinger (DQS) equation, which formally describes the behavior of small solutions of the MRS system on order $O(\frac{1}{\epsilon^2})$ timescale. The degeneracy of the equation makes it very difficult to obtain well-posedness results. In fact, previous work by Jeong and Oh has shown the DQS equation is ill-posed in $H^s$ for sufficiently large $s$. However, we will show that the equation is well-posed in a highly restricted function space of compactly supported solutions with sufficient endpoint decay.

Numerical simulations of the DQS equation appear to confirm that it does provide a good asymptotic description of the behavior of the MRS solutions on timescale $O(\frac{1}{\epsilon^2})$. In addition, we study the possible existence of compactly supported solutions through numerical simulations. The numerical results of a front spreading solution seem to suggest that, in the absence of dissipation, a compactly supported initial pulse forms oscillations that spread to $\pm \infty$ once a singularity occurs, and therefore, does not stay compact for later time.