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1
Content available remote

The Cauchy problem and self-similar solutions for a nonlinear parabolic equation

100%
Biler P.
Studia Mathematica
|
1995
|
tom 114
|
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.
2
Content available remote

A note on the paper of Y. Naito

100%
Biler P.
Banach Center Publications
|
2006
|
tom 74
|
nr 1
33-40
EN
This note contains some remarks on the paper of Y. Naito concerning the parabolic system of chemotaxis and published in this volume.
3
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Radially symmetric solutions of a chemotaxis model in the plane-the supercritical case

100%
Biler P.
Banach Center Publications
|
2008
|
tom 81
|
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
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Growth and accretion of mass in an astrophysical model

100%
Biler P.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 2
179-189
EN
We study asymptotic behavior of radial solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles. In particular, we consider stationary solutions in balls and in the whole space, self-similar solutions defined globally in time, blowing up self-similar solutions, and singularities of solutions that blow up in a finite time.
5
Content available remote

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

100%
Biler P.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
229-239
6
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Growth and accretion of mass in an astrophysical model, II

64%
Biler P., Nadzieja T.
Applicationes Mathematicae
|
1995-1996
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tom 23
|
nr 3
351-361
EN
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.
7
Content available remote

Global existence versus blow up for some models of interacting particles

64%
Biler P., Brandolese L.
Colloquium Mathematicum
|
2006
|
tom 106
|
nr 2
293-303
EN
We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and Debye-Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method due to S. Montgomery-Smith.
8
Content available remote

Existence of Solutions for the Keller-Segel Model of Chemotaxis with Measures as Initial Data

64%
Biler P., Zienkiewicz J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 1
41-51
EN
A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than 8π as the initial data is given. This result was obtained by Senba and Suzuki (2002) and Bedrossian and Masmoudi (2014) using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.
9
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Nonlocal quadratic evolution problems

64%
Biler P., Woyczyński W. A.
Banach Center Publications
|
2000
|
tom 52
|
nr 1
11-24
EN
Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data. Monte Carlo approximation schemes by interacting particle systems are also mentioned.
10
Content available remote

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

64%
Biler P., Nadzieja T.
Colloquium Mathematicum
|
1993
|
tom 66
|
nr 2
319-334
EN
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.
11
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A Neumann problem for a convection-diffusion equation on the half-line

64%
Biler P., Karch G.
Annales Polonici Mathematici
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2000
|
tom 74
|
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.
12
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A class of nonlocal parabolic problems occurring in statistical mechanics

64%
Biler P., Nadzieja T.
Colloquium Mathematicum
|
1993
|
tom 66
|
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.
13
Content available remote

An elementary approach to nonexistence of solutions of linear parabolic equations

64%
Biler P., Nadzieja T.
Colloquium Mathematicum
|
2011
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tom 122
|
nr 1
125-134
EN
This note presents an elementary approach to the nonexistence of solutions of linear parabolic initial-boundary value problems considered in the Feller test.
14
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On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis

64%
Biler P., Brandolese L.
Studia Mathematica
|
2009
|
tom 193
|
nr 3
241-261
EN
We establish new results on convergence, in strong topologies, of solutions of the parabolic-parabolic Keller-Segel system in the plane to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.
15
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Generalized Fokker-Planck equations and convergence to their equilibria

64%
Biler P., Karch G.
Banach Center Publications
|
2003
|
tom 60
|
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.
16
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Asymptotics for conservation laws involving Lévy diffusion generators

52%
Biler P., Karch G., Woyczyński W.
Studia Mathematica
|
2001
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tom 148
|
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.
17
Content available remote

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

52%
Biler P., Hilhorst D., Nadzieja T.
Colloquium Mathematicum
|
1994
|
tom 67
|
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.
18
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Asymptotics for multifractal conservation laws

52%
Biler P., Karch G., Woyczynski W. A.
Studia Mathematica
|
1999
|
tom 135
|
nr 3
231-252
EN
We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t + f(u)_x = Au$, where the operator A is a linear combination of fractional powers of the second derivative $-∂^2/ ∂ x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.
19
Content available remote

Asymptotic stability of Navier-Stokes flow past an obstacle

52%
Biler P., Cannone M., Karch G.
Banach Center Publications
|
2004
|
tom 66
|
nr 1
47-59
EN
Results on the asymptotic stability of solutions of the exterior Navier-Stokes problem in ℝ³ are proved in the framework of weak $L^p$ spaces.
20
Content available remote

Nonisothermal systems of self-attracting Fermi-Dirac particles

52%
Biler P., Nadzieja T., Stańczy R.
Banach Center Publications
|
2004
|
tom 66
|
nr 1
61-78
EN
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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