Unipolar Euler–Poisson equations with time-dependent damping: blow-up and global existence

Abstract

This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler–Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of $-\dfrac{\mu}{(1+t)^\lambda} \rho \mu$ for $\lambda \gt 0$ with $\mu \gt 0$ plays a crucial role for the structure of solutions. The main issue of the paper is to investigate the critical case with $\lambda=1$. We first prove that, for all cases with $\lambda \gt 0$ and $\mu \gt 0$ (including the critical case of $\lambda=1$), once the initial data is steep at a point, then the solutions are locally bounded but their derivatives will blow up in finite time, by means of the method of Riemann invariants and the technical convex analysis. Secondly, for the critical case of $\lambda=1$ with $\mu \gt 7/3$, we prove that there exists a unique global solution, once the initial perturbation around the constant steady-state is sufficiently small. In particular, we derive the algebraic convergence rates of the solution to the constant steady-state, which are piecewise, related to the parameter $\mu$ for $7/3 \lt \mu \leq 3$, $3 \lt \mu \leq 4$ and $\mu \gt 4$. The adopted method of proof in this critical case is the technical time-weighted energy method and the time-weight depends on the parameter $\mu$. Finally, we carry out some numerical simulations in two cases for blow-up and global existence, respectively, which numerically confirm our theoretical results.