The present paper deals with the critical threshold phenomena for a one-dimensional (1D) damped, pressureless Euler-Poisson system, with potential (\(\phi\)) induced by a constant background (\(c>0\)), namely \(\rho_t +(\rho u)_x =0\); \(u_t +uu_x=-k\phi_x-\nu u\); \(-\phi_{xx} = \rho-c\) in \(\mathbb{R}\times (0,T)\), subject to initial data \((\rho_0,u_0)\in [C^1(\mathbb{R})]^2\), with \(\rho_0>0\) obeying the neutrality condition \(\int_{-\infty}^{+\infty} (\rho_0(\xi) -c)\,\mathrm{d}\xi=0\), originally studied in [S. Engelberg et al., Indiana Univ. Math. J. 50, Spec. Iss., 109–157 (2001; Zbl 0989.35110)]. Here, \(\nu>0\) denotes the damping coefficient, and the parameter \(k>0\) represents the repulsive behavior of the underlying forcing. The authors present two results. In the first one, they establish necessary and sufficient conditions on the initial data to the existence of the finite time breakdown of the \(C^1\) solution to the problem under study, for three damping cases:

(I)

strong damping (\(\nu> 2\sqrt{kc}\));

(II)

borderline damping (\(\nu= 2\sqrt{kc}\));

(III)

weak damping (\(\nu< 2\sqrt{kc}\)).

In the second one, the authors establish necessary and sufficient conditions on the initial data to the existence of a unique global solution. The initial steps in both proofs are:

(1)

to convert the PDE system to a nonlinear ODE system along the particle path, which is fixed for a fixed value of the parameter \(\alpha = x(0)\), by employing the method of characteristics and letting \(d =u_x\);

(2)

and to reduce the nonlinear ODE system to a linear ODE system, by transforming the variables, more precisely \(r=-d/\rho\) and \(s= 1/\rho\).

The proof of the first result relies on the analysis of the unique explicit solution of the IVP, \(s''+\nu s'+ kc s =k\), \(s(0)=1/\rho_0\) and \(s''(0) = d_0/\rho_0\), which is derived from the linear ODE system. The proof of the second result relies on the geometric interpretation in terms of phase (\(r-s\)) plane analysis. Finally, the authors apply the obtained results to identify the critical thresholds in a 1D Euler system in the context of biological aggregation.