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1
Content available remote Existence and nonexistence of solutions for a model of gravitational interaction of particles, III
100%
Biler P.
|
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nr 2
229-239
2
Content available remote The Cauchy problem and self-similar solutions for a nonlinear parabolic equation
100%
Biler P.
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nr 2
181-205
EN
The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
3
Content available remote Radially symmetric solutions of a chemotaxis model in the plane-the supercritical case
100%
Biler P.
Banach Center Publications
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2008
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tom 81
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nr 1
31-42
EN
The existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane ℝ² are studied for the (supercritical) value of mass greater than 8π.
4
Content available remote A Neumann problem for a convection-diffusion equation on the half-line
63%
Biler P. , Karch G.
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2000
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tom 74
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nr 1
79-95
EN
We study solutions to a nonlinear parabolic convection-diffusion equation on the half-line with the Neumann condition at x=0. The analysis is based on the properties of self-similar solutions to that problem.
5
Content available remote Generalized Fokker-Planck equations and convergence to their equilibria
63%
Biler P. , Karch G.
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tom 60
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nr 1
307-318
EN
We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on $ℝ^{d}$ with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.
6
Content available remote A class of nonlocal parabolic problems occurring in statistical mechanics
63%
Biler P. , Nadzieja T.
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tom 66
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nr 1
131-145
EN
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.
7
Content available remote Asymptotics for conservation laws involving Lévy diffusion generators
51%
Biler P. , Karch G. , Woyczyński W.
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nr 2
171-192
EN
Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their $L^{p}$-decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.
8
Content available remote Existence and nonexistence of solutions for a model of gravitational interaction of particles, II
51%
Biler P. , Hilhorst D. , Nadzieja T.
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nr 2
297-308
EN
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.
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