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.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

scholarly journals Critical exponents for a semilinear parabolic equation with variable reaction

2012 ◽  
Vol 142 (5) ◽  
pp. 1027-1042 ◽  
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
M. Pérez-LLanos ◽  
J. D. Rossi
Keyword(s):  
Critical Exponents ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Parabolic Problem ◽  
Blow Up ◽  
Bounded Function ◽  
Semilinear Parabolic Equation ◽  
All Solutions ◽  
Image Position ◽  
Semilinear Parabolic

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.When Ω = ℝN we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

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.

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

Thermal runaway in a non-local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway

1995 ◽  
Vol 6 (3) ◽  
pp. 201-224 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Steady State ◽  
Blow Up ◽  
Ohmic Heating ◽  
Local Problem ◽  
Asymptotically Stable ◽  
Local Behaviour ◽  
Globally Asymptotically Stable ◽  
Non Local ◽  
Image Position ◽  
Unique Steady State

We consider the non-local problemIt is found that for the case of decreasingfthen: (i) forthere is a unique steady state which is globally asymptotically stable; (ii) forthen the problem can be scaled so thatin which case: (a) for λ < 8 there is a unique steady state which is globally asymptotically stable; (b) for λ = 8 there is no steady state anduis unbounded; (c) for λ > 8 there is no steady state andublows up for allx, −1 <x, < 1. Some formal asymptotic estimates for the local behaviour ofuas it blows up are obtained.

scholarly journals BLOWUP FOR A DEGENERATE AND SINGULAR PARABOLIC EQUATION WITH NON-LOCAL SOURCE AND ABSORPTION

2009 ◽  
Vol 52 (2) ◽  
pp. 209-225 ◽  
Author(s):  
JUN ZHOU ◽  
CHUNLAI MU
Keyword(s):  
Parabolic Equation ◽  
Finite Time ◽  
Sufficient Conditions ◽  
Singular Equation ◽  
Local Source ◽  
Negative Solution ◽  
Singular Parabolic Equation ◽  
Non Local ◽  
Image Position

AbstractThis paper deals with the following degenerate and singular equation with non-local source and absorption. The existence of a unique classical non-negative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained.

scholarly journals Finite-time blow-up of a non-local stochastic parabolic problem

2020 ◽  
Vol 130 (9) ◽  
pp. 5605-5635
Author(s):  
Nikos I. Kavallaris ◽  
Yubin Yan
Keyword(s):  
Finite Time ◽  
Parabolic Problem ◽  
Blow Up ◽  
Non Local

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

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