Long-time behaviours of classical solutions to relativistic Euler–Poisson equations

Abstract

In this paper, long-time behaviours of classical solutions to relativistic Euler–Poisson equations in radial symmetry (REPE) are investigated. First, we establish the finite propagation speed property of the REPE system and the propagation speed $\sigma$ is found to be

$$\begin{aligned} \sigma :=\sqrt{\frac{8\pi M}{R(1+e/c^{2})}+c^2}, \end{aligned}$$

where all constants are given parameters or initial data to the system. Subsequently, we show that if the initial functional

$$\begin{aligned} \int \limits _0^{R}v(0,r)\textrm{d}r \end{aligned}$$

is sufficiently large, then the classical solutions of the REPE will blow up on finite time, where v(0, r) is the initial velocity component of REPE and R is the radius of support of the initial mass-energy density.

Appendix

In this appendix, we provide an independent result which describes how a type of general first-order differential inequality will blow up on finite time.

Lemma 13

Consider the following differential inequality:

$$\begin{aligned} {H}'(t)\ge \frac{H^2(t)}{A(t)}-G(t)\quad t\ge t_0 \end{aligned}$$

(107)

such that A and G are given positive and increasing functions of $t\ge t_0$. We have, for any $\tau > t_0$, if

$$\begin{aligned} H(t_0)>C(\tau ), \end{aligned}$$

(108)

then the solutions blow up on or before $\tau$, where $C(\tau )$ is the maximum of

$$\begin{aligned} \sqrt{2A(\tau )G(\tau )} \end{aligned}$$

(109)

and

$$\begin{aligned} \left[ \int \limits _{t_0}^\tau \frac{1}{2A(s)}ds\right] ^{-1}. \end{aligned}$$

(110)

Proof

Fix $\tau >0$, for any $t_0\le t\le \tau$, (107) becomes

$$\begin{aligned} {H}'(t)&\ge \frac{H^2(t)}{A(t)}-G(t)\end{aligned}$$

(111)

$$\begin{aligned}&=\frac{H^2(t)}{2A(t)}+\left[ \frac{H^2(t)}{2A(t)}-G(t)\right] \end{aligned}$$

(112)

$$\begin{aligned}&\ge \frac{H^2(t)}{2A(t)}+\left[ \frac{H^2(t)}{2A(\tau )}-G(\tau )\right] \end{aligned}$$

(113)

$$\begin{aligned}&=:\frac{H^2(t)}{2A(t)}+Q(t). \end{aligned}$$

(114)

By (108), $Q(t_0)>0$. Hence $Q(t)\ge 0$ on $[t_0,\tau ]$ and

$$\begin{aligned} {H}'(t)\ge \frac{H^2(t)}{2A(t)} \end{aligned}$$

(115)

on $[t_0,\tau ]$. Therefore,

$$\begin{aligned} \frac{1}{H(t_0)}-\frac{1}{H(t)}&\ge \int \limits _{t_0}^t\frac{1}{2A(s)}\textrm{d}s\end{aligned}$$

(116)

$$\begin{aligned} 0<\frac{1}{H(t)}&\le \frac{1}{H(t_0)}-\int \limits _{t_0}^t\frac{1}{2A(s)}\textrm{d}s. \end{aligned}$$

(117)

By (108), the right-hand side of (117) is negative when $t=\tau$. Thus, the solutions blow up on or before time $\tau$. The proof is completed. $\square$