Singularity formation for the compressible non-isentropic Euler equations with time-dependent damping

Abstract

In this paper, we mainly study the blow up phenomenon to classical solutions of compressible non-isentropic Euler equations with time-dependent damping $\frac{a}{{(1+t)}^{\lambda }}u$ in one space dimension. By constructing the decoupled Riccati equation, we show that $C^1$ solutions will blow up in finite time when the adiabatic gas constant $1<\gamma <3$ and the damping coefficient $\lambda \ge 0$ if the initial data satisfies suitable condition. Moreover, when the initial data is small enough, we can see that the blow up comes from derivatives of the solution.