scholarly journals SOME INTERESTING SPECIAL CASES OF A NON-LOCAL PROBLEM MODELLING OHMIC HEATING WITH VARIABLE THERMAL CONDUCTIVITY

2001 ◽  
Vol 44 (3) ◽  
pp. 585-595 ◽  
Author(s):  
D. E. Tzanetis ◽  
P. M. Vlamos
Keyword(s):  
Stationary Solution ◽  
Dirichlet Boundary ◽  
Ohmic Heating ◽  
Variable Thermal Conductivity ◽  
Dirichlet Boundary Conditions ◽  
Heaviside Function ◽  
Local Equation ◽  
Special Cases ◽  
Primary 35B30 ◽  
Non Local

AbstractThe non-local equation$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

Asymptotic formulas for Dirichlet boundary value problems

2005 ◽  
Vol 42 (2) ◽  
pp. 153-171 ◽  
Author(s):  
Bülent Yilmaz ◽  
O. A. Veliev
Keyword(s):  
Boundary Conditions ◽  
Main Part ◽  
Dirichlet Boundary ◽  
Arbitrary Order ◽  
Dirichlet Boundary Conditions ◽  
Asymptotic Formulas ◽  
Summable Function ◽  
Special Cases ◽  
Sturm Liouville ◽  
Liouville Operators

In this article we obtain asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the nonselfadjoint Sturm-Liouville operators with Dirichlet boundary conditions, when the potential is a summable function. Then using these we compute the main part of the eigenvalues in special cases.

Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some special cases

1995 ◽  
Vol 6 (2) ◽  
pp. 127-144 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Steady State ◽  
Ohmic Heating ◽  
Local Problem ◽  
Temperature Dependent ◽  
Globally Asymptotically Stable ◽  
Special Cases ◽  
Non Local ◽  
Model Derivation ◽  
Current Flows ◽  
Image Position

We consider the non-local problemwhich models the temperature when an electric current flows through a material with temperature dependent electrical resistivity f(u) > 0, subject to a fixed potential difference. It is found that for some special cases where f is decreasing andso the problem can be scaled to makethen:(a) for λ < 8 there is a unique steady state which is globally asymptotically stable: (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, – 1 < x < 1.

Asymptotic behaviour for a non-local parabolic problem

2009 ◽  
Vol 20 (3) ◽  
pp. 247-267 ◽  
Author(s):  
LIU QILIN ◽  
LIANG FEI ◽  
LI YUXIANG
Keyword(s):  
Asymptotic Behaviour ◽  
Stationary Solution ◽  
Finite Time ◽  
Parabolic Problem ◽  
Dirichlet Boundary ◽  
Asymptotic Estimates ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Globally Asymptotically Stable ◽  
Non Local ◽  
Image Position

In this paper, we consider the asymptotic behaviour for the non-local parabolic problemwith a homogeneous Dirichlet boundary condition, where λ > 0,p> 0 andfis non-increasing. It is found that (a) for 0 <p≤ 1,u(x,t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 <p< 2,u(x,t) is globally bounded for any λ > 0; (c) forp= 2, if 0 < λ < 2|∂Ω|2, thenu(x,t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution andu(x,t) is a global solution andu(x,t) → ∞ ast→ ∞ for allx∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution andu(x,t) blows up in finite time for allx∈ Ω; (d) forp> 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* andu0(x) sufficiently large,u(x,t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour ofu(x,t) as it blows up are obtained forp≥ 2.

Monotone iterative sequences for non-local elliptic problems

2011 ◽  
Vol 22 (6) ◽  
pp. 533-552 ◽  
Author(s):  
MOHAMMED AL-REFAI ◽  
NIKOS I. KAVALLARIS ◽  
MOHAMED ALI HAJJI
Keyword(s):  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Iterative Sequence ◽  
Monotone Sequence ◽  
Monotone Iterative ◽  
Elliptic Differential Equations ◽  
Actual Solution ◽  
Non Local

In this paper we establish an existence and uniqueness result for a class of non-local elliptic differential equations with the Dirichlet boundary conditions, which, in general, do not accept a maximum principle. We introduce one monotone sequence of lower–upper pairs of solutions and prove uniform convergence of that sequence to the actual solution of the problem, which is the unique solution for some range of λ (the parameter of the problem). The convergence of the iterative sequence is tested through examples with an order of convergence greater than 1.

scholarly journals Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Nikos I. Kavallaris ◽  
Raquel Barreira ◽  
Anotida Madzvamuse
Keyword(s):  
Stationary Solution ◽  
Analytical Methods ◽  
Blow Up ◽  
Turing Instability ◽  
Numerical Approach ◽  
Local Equation ◽  
Shadow System ◽  
Non Local ◽  
Stability Results ◽  
Theoretical Results

AbstractThe main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

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