### A model of a radially symmetric cloud of self-attracting particles

We consider a parabolic equation which describes the gravitational interaction of particles. Existence of solutions and their convergence to stationary states are studied.

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We consider a parabolic equation which describes the gravitational interaction of particles. Existence of solutions and their convergence to stationary states are studied.

Need to develop applications of mathematics meets with acceptance, understanding and sympathetic interest in the mathematicians world. Evidenced are by creation of the mathematics major with ... or the mathematics in ..., adding in a purely mathematical work of a few sentences or even a chapter on the possible applications of the results, conference papers often are preceded by an introductory discussion the motivations of the biological, physical or chemical character to deal with the issue. Organized...

We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.

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.

Some conditions for the existence and uniqueness of solutions of the nonlocal elliptic problem $-\Delta \phi =Mf\left(\phi \right)/\left({\left({\int}_{\Omega}f\left(\phi \right)\right)}^{p}\right)$, ${\phi |}_{\Omega}=0$ are given.

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.

We consider the following problem: $\Delta \Phi =\pm Mover{\int}_{\Omega}{e}^{-\Phi /\Theta}{e}^{-\Phi /\Theta},E=M\Theta \mp 1over2{\int}_{\Omega}{\left|\nabla \Phi \right|}^{2}{,\Phi |}_{\partial \Omega}=0,$ where Φ: Ω ⊂ ${\mathbb{R}}^{n}$ → ℝ is an unknown function, Θ is an unknown constant and M, E are given parameters.

Existence of radially symmetric solutions (both stationary and time dependent) for a parabolic-elliptic system describing the evolution of the spatial density of ions in an electrolyte is studied.

We study the existence and uniqueness of the steady state in a model describing the evolution of density of bacteria and oxygen dissolved in water filling a capillary. The steady state is a stationary solution of a nonlinear and nonlocal problem which depends on the energy function and contains two parameters: the total mass of the colony of bacteria and the concentration (or flux) of oxygen at the end of the capillary. The existence and uniqueness of solutions depend on relations between these...

We consider a model of migrating population occupying a compact domain Ω in the plane. We assume the Malthusian growth of the population at each point x ∈ Ω and that the mobility of individuals depends on x ∈ Ω. The evolution of the probability density u(x,t) that a randomly chosen individual occupies x ∈ Ω at time t is described by the nonlocal linear equation ${u}_{t}={\int}_{\Omega}\phi \left(y\right)u(y,t)dy-\phi \left(x\right)u(x,t)$, where φ(x) is a given function characterizing the mobility of individuals living at x. We show that the asymptotic behaviour of u(x,t)...

This note presents an elementary approach to the nonexistence of solutions of linear parabolic initial-boundary value problems considered in the Feller test.

We study convergence of solutions to stationary states in an astrophysical model of evolution of clouds of self-gravitating particles.

We study the existence and nonexistence in the large of radial solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles. The blow-up of solutions defined in the n-dimensional ball with large initial data is connected with the nonexistence of radial stationary solutions with a large mass.

The existence of stationary solutions and blow up of solutions for a system describing the interaction of gravitationally attracting particles that obey the Fermi-Dirac statistics are studied.

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