Growth and accretion of mass in an astrophysical model, II

Abstract

Abstract. Radially symmetric solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles in a bounded container are studied. Conditions ensuring either global-in-time existence of solutions or their finite time blow up are given.1. Introduction. In the second part of [1] we continue the study of radially symmetric solutions to the parabolic-elliptic system considered in (1)Among physical interpretations of the system (1)- (2) we cite the evolution version of the Chandrasekhar equation from the theory of gravitating stars. For a discussion of other motivations leading to the nonlocal parabolic equation (1) of Fokker-Planck type we refer the reader to the introductions in [5], [10] and [11][12]. A related system of two parabolic equations modelling a biological phenomenon of chemotaxis has been studied in [7]. When the system (1)- (2) is considered in a bounded domain Ω ⊂ R n , the nonlinear no-flux condition for the density of particles u,is a natural one, since it guarantees the conservation of the total mass M = Ω u(x, t) dx.

A Singular Problem in Electrolytes Theory

“…For q*'4 and q '0, there is no solution of (10)- (11). Similarly, the stationary equation (10) does not admit any nontrivial positive solution Q, Q(0)"0, in the whole plane 1.…”

“…Let us begin with the radial case, which has been studied for related parabolicelliptic systems in [6,Appendix]; [5], [7], [10]. The essential difficulty is the singular diffusion coefficient 4y in (10) for n"2.…”

Asymptotic self-similar blow-up for a model of aggregation

“…Proof of Corollary 2.2. If b 0 ≡ 1 and Θ < 1/(2(d + 2)χ d ), we know from [7,Theorem 2] that the corresponding solution b blows up. Now assuming Θ ≤ 1/(4dχ d ) < 1/(2(d + 2)χ d ), we can apply Theorem 2.1 to conclude.…”

“…If T * < ∞, then we say that blow-up occurs for (1)- (6), more precisely, one finds lim t→T * sup [0,1] n(r, t) = ∞. (8) There are various conditions in the literature which ensure T * < ∞ [3,4,7,6]. For instance, in [4] it was found for the radially symmetric case that T * < ∞ holds for any bounded initial condition and Θ < 1/(2d 2 χ d ).…”

Grow-Up Rate and Refined Asymptotics for a Two-Dimensional Patlak–Keller–Segel Model in a Disk

“…Moreover a "gradient blow-up" occurs at t = T * , in the sense that N x (0, t) → ∞ as t → T * and the density ρ becomes also unbounded at time T * , cf. [8,Theorem 2(i)]. On the other hand, for 0 < Λ < 8π and any (admissible) initial data there is a unique global-in-time solution N ∈ C([0, ∞); L 2 (0, 1)) ∩ C 2,1 ((0, 1) × (0, ∞)).…”