Existence and nonexistence of solutions for a model of gravitational interaction of particles. I. (English) Zbl 0817.35041

Existence properties are studied for the parabolic-elliptic system \[ (1) \quad u_ t= \Delta u+ \nabla\cdot (u\nabla \varphi) \quad \text{in } (0,T)\times \Omega, \qquad (2) \quad \Delta\varphi= u\quad \text{in } \Omega \] subject to the nonlinear no-flux condition \[ {{\partial u} \over {\partial \nu}}+u {{\partial\varphi} \over {\partial\nu}} =0 \qquad \text{on } (0,T)\times \partial\Omega \tag{3} \] (\(\nu\) is the unit outward normal to \(\partial\Omega\)). It is assumed that the potential \(\varphi\) satisfies either (4.1) \(\varphi=0\) on \(\partial\Omega\) or, (4.2) \(\varphi= E_ n*u\), where \(E_ n\) is the fundamental solution of the \(n\)-dimensional Laplacian. Finally, the initial condition (5) \(u(0,x)= u_ 0 (x)\geq 0\) is imposed.
The authors study existence, nonexistence and uniqueness of the stationary solution to the system (1)–(4). Further, local- and global- in-time existence results are proved for the evolutional system (1)–(5). It is shown that for star-shaped domains \(\Omega\) and large initial data the solutions of (1)–(5) blow-up in a finite time. The problem under consideration is an evolution version of the Chandrasekhar equation and it describes the gravitational interaction of particles.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
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