# Singularity formation for the 1D compressible Euler equation with variable damping coefficient

## TL;DR

This paper studies conditions under which solutions to the 1D compressible Euler equation with variable damping blow up in finite time, providing criteria based on initial data and damping decay rates.

## Contribution

It introduces new sufficient conditions for finite-time blow-up in the 1D Euler equation with variable damping, using Riemann invariants and lifespan estimates.

## Key findings

- Finite-time blow-up occurs under certain damping decay conditions.
- Derived sharp lifespan estimates for small perturbations.
- Bounded solutions with derivative blow-up are characterized.

## Abstract

In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial data are small perturbations of constant states.