Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

TL;DR

This paper investigates the 1D damped compressible Euler equations with variable damping, establishing conditions for global existence or blow-up of solutions based on the damping coefficient's properties.

Contribution

It extends known results by proving invariance of global existence and blow-up criteria under time and space dependent damping perturbations.

Findings

Global existence for  with <

Finite-time blow-up for =0

Conditions on damping coefficient for solution behavior

Abstract

In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or $0$, the solution exists globally in time or blows up in finite time, respectively. We prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient $a$ satisfies the following condition $$ |a(t,x)- \mu_0| \leq a_1(t) + a_2 (x), $$ where $\mu_0 \geq 0$ and $a_1$ and $a_2$ are integrable functions with $t$ and $x$. Under this condition, we show the global existence and the blow-up with small initial data, when $\mu_0 >0$ and $\mu=0$ respectively.