Some results concerning the Poisson-Boltzmann equation

Krzywicki, A.; Nadzieja, Tadeusz

doi:10.4064/am-21-2-265-272

Cited by 20 publications

(15 citation statements)

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“…Lions [26] showed the existence of a unique regular solution of problem (10)- (11) in the electrostatic case in the space H 1 o (Ω). In the gravitational case the PB equation has been studied by A. Krzywicki and T. Nadzieja in [28,29]. They proved that problem (10)-(11) has no classical solutions for α > 0 large enough for simply connected domains.…”

Section: The Last Two Terms Are Zero Because Of Mass Conservation Andmentioning

confidence: 99%

“…On the other hand, if Ω is an annulus, (35) has at least a solution for any value of the parameter α (see [28]). …”

Section: Large-time Asymptoticsmentioning

confidence: 99%

“…u Ω e u dx (35) with Dirichlet boundary conditions, where α = β σ M . Equation (35) has solution(s) depending strongly on the topology of the domain Ω (see [28,29]). In fact, if Ω is the unit ball B(0, 1), we have the following properties:…”

Section: Large-time Asymptoticsmentioning

confidence: 99%

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Asymptotic Behavior of an Initial-Boundary Value Problem for the Vlasov--Poisson--Fokker--Planck System

Bonilla¹,

Carrillo²,

Soler³

1997

SIAM J. Appl. Math.

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Abstract. The asymptotic behavior for the Vlasov-Poisson-Fokker-Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson-Boltzmann equation with Dirichlet boundary condition. In the proof of the main result, the conservation law of mass and the balance of energy and entropy identities are rigorously derived. An important argument in the proof is to use a Lyapunov-type functional related to these physical quantities.

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Section: The Last Two Terms Are Zero Because Of Mass Conservation Andmentioning

confidence: 99%

“…On the other hand, if Ω is an annulus, (35) has at least a solution for any value of the parameter α (see [28]). …”

Section: Large-time Asymptoticsmentioning

confidence: 99%

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Asymptotic Behavior of an Initial-Boundary Value Problem for the Vlasov--Poisson--Fokker--Planck System

Bonilla¹,

Carrillo²,

Soler³

1997

SIAM J. Appl. Math.

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“…Thus 19) and for any fixed x ∈ Ω the second term on the right-hand side of (4.19) is much smaller than the first as t → t * , due to the contribution of the Green function term, and so…”

Section: (See Also the Comments After The Definition Of V(x; µ ν δ)mentioning

confidence: 99%

“…In the case of a nonlinear conductor, when |j| ∝ |∇φ| k , k > 0, a two-dimensional version 5) p > 1, can be derived to describe the thermoelectric flow in the conductor (see [22] and the references therein). Also, for p = 1 equation (1.5) can describe phenomena associated with the occurrence of shear bands in metals being deformed under high strain rates [6][7][8], in the theory of gravitational equilibrium of polytropic stars [19], in the investigation of the fully turbulent behaviour of real flows, using invariant measures for the Euler equation [9], and in modelling aggregation of cells via interaction with a chemical substance (chemotaxis) [24].…”

Section: Introductionmentioning

confidence: 99%

On the Blow-Up of the Non-Local Thermistor Problem

Kavallaris

Nadzieja²

2007

Proceedings of the Edinburgh Mathematical Society

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The conditions under which the solution of the non-local thermistor problemblows up are investigated. We assume that f (s) is a decreasing function and that it is integrable in (0, ∞). Considering a suitable functional we prove that for all λ > 0 the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that λ is sufficiently large (λ 1). In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either λ > λ * or 0 < λ < λ * , for some critical value λ * , and that the initial condition is sufficiently large provided also that f (s) satisfies the decay condition

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Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

Latos

Tzanetis

2009

Nonlinear Differ. Equ. Appl.

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We investigate the behaviour of solution u = u(x, t; λ) at λ = λ * for the non-local porous medium equation ut = (u n )xx + λf (u)/ ( 1 −1 f (u)dx) 2 with Dirichlet boundary conditions and positive initial data. The function f satisfies: f (s), −f (s) > 0 for s ≥ 0 and s n−1 f (s) is integrable at infinity. Due to the conditions on f , there exists a critical value of parameter λ, say λ * , such that for λ > λ * the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ * the corresponding steady-state problem does not have any solution. For 0 < λ < λ * there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u * = u(x, t; λ * ) (u * (x, t) → ∞, as t → ∞ for all x ∈ (−1, 1)). Mathematics Subject Classification (2000). Primary 35K55; Secondary 35B05.

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Some results concerning the Poisson-Boltzmann equation

Cited by 20 publications

References 0 publications

Asymptotic Behavior of an Initial-Boundary Value Problem for the Vlasov--Poisson--Fokker--Planck System

Asymptotic Behavior of an Initial-Boundary Value Problem for the Vlasov--Poisson--Fokker--Planck System

On the Blow-Up of the Non-Local Thermistor Problem

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

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