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

Some personal remarks about applications of mathematics

Potrzeba rozwijania zastosowań matematyki spotyka się z akceptacją, zrozumieniem i życzliwym zainteresowaniem w środowisku matematyków. Świadczą o tym tworzone specjalizacje matematyka z ... lub zastosowania matematyki w ..., dopisywanie w pracach czysto matematycznych paru zdań lub nawet rozdziału o możliwych zastosowaniach uzyskanych wyników, często też referaty konferencyjne poprzedzone są wstępem omawiającym motywacje biologiczne, fizyczne czy chemiczne zajmowania się danym zagadnieniem. Organizuje się też wiele konferencji matematycznych poświęconych zastosowaniom matematyki w różnych dziedzinach nauk przyrodniczych lub ekonomicznych. Wszystko to daje fałszywy obraz bujnie rozwijających się zastosowań matematyki i ekspansji metod matematycznych w różnych dziedzinach nauk.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 a number of conferences devoted to mathematical applications of mathematics in various fields of natural sciences or economics. All this gives a false picture of lush developing applications of mathematics and expansion of mathematical methods in various fields of science

Nonlocal elliptic problems

Some conditions for the existence and uniqueness of solutions of the nonlocal elliptic problem $-Δφ = M f(φ)/((∫_{Ω} f(φ))^p)$, $φ|_{\partial Ω}=0$ are given

Radially symmetric solutions of the Poisson-Boltzmann equation with a given energy

We consider the following problem: $ΔΦ = ± {M οver \int_{Ω} e^{- Φ/Θ}} e^{- Φ/Θ}, E = MΘ ∓ {1οver2}\int_{Ω} |∇Φ|^2, Φ|_{\partial Ω} = 0,$ where Φ: Ω ⊂ $ℝ^n$ → ℝ is an unknown function, Θ is an unknown constant and M, E are given parameters

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

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

A singular radially symmetric problem in electrolytes theory

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

Growth and accretion of mass in an astrophysical model, II

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

A class of nonlocal parabolic problems occurring in statistical mechanics

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

Structure of steady states for Streater's energy-transport models of gravitating particles

Energy-transport models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study the existence of steady states with prescribed mass and energy for these models

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

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