Existence and nonexistence of solutions for a model of gravitational interaction of particles, I

Abstract

We study the existence of stationary and evolution solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles.

Growth and accretion of mass in an astrophysical model, II

“…quenching for (10). This gives an insight into the mechanism of formation of singularities leading to nonglobal existence of radial solutions proved in [4,Th. 3].…”

“…Introduction. In the second part of [1] we continue the study of radially symmetric solutions to the parabolic-elliptic system considered in [3][4][5], [10]: (1) u t = ∆u + ∇ · (u∇ϕ),…”

“…We remark that Theorem 2 in [4] deals with Q which are classical solutions of (10)- (11) in (0, 1)×(0, ∞), and may correspond to unbounded u, while here Q will be smooth in [0, 1] × (0, ∞). Moreover, we will prove finite time blow up of solutions for certain Q 0 (Theorem 2), a phenomenon described by an unbounded growth of the derivative ∂ ∂y Q, i.e.…”

“…The idea of constructing suitable super-and subsolutions was suggested by the form of stationary solutions in [4,Sec. 4], by (9) written as…”

“…(iv) We recall that for n ≥ 5, (10)- (11) The convergence of solutions as t → +∞ in Theorem 1(ii), (iv) can be obtained by using a similar argument involving Lyapunov functions introduced in [4][5]. P r o o f o f T h e o r e m 1.…”