The lifespan of classical solutions to the (damped) compressible Euler equations

In this paper, the initial-boundary value problem of the original three-dimensional compressible Euler equations with (or without) time-dependent damping is considered. By considering a functional F(t,α,f) weighted by a general time-dependent parameter function α and a general radius-dependent parameter function f, we show that if the initial value F|_t=0 is sufficiently large, then the lifespan of the system is finite. Here, f can be any C¹ strictly increasing function such that the sum of initial values of f and α is non-negative. It follows that a class of conditions for non-existence of global classical solutions is established. Moreover, the conditions imply that a strong α will lead to a more unrestrained necessary condition for classical solutions of the system to exist globally in time. Copyright © 2020 Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia.