The limiting Behavior of Solutions to Inhomogeneous Eigenvalue Problems in Orlicz-Sobolev Spaces

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Marian Bocea ◽  
Mihai Mihăilescu ◽  
Denisa Stancu-Dumitru
Keyword(s):  
Asymptotic Behavior ◽  
Sobolev Spaces ◽  
Eigenvalue Problems ◽  
Limiting Behavior

AbstractThe asymptotic behavior of the sequence {u

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

2018 ◽  
Vol 111 (1) ◽  
pp. 43-68
Author(s):  
Sergey Shmarev ◽  
Jacson Simsen ◽  
Mariza Stefanello Simsen ◽  
Marcos Roberto T. Primo
Keyword(s):  
Asymptotic Behavior ◽  
Sobolev Spaces ◽  
Parabolic Equations ◽  
Weighted Variable

scholarly journals Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

2019 ◽  
Vol 29 (08) ◽  
pp. 1585-1617 ◽  
Author(s):  
Yvon Maday ◽  
Carlo Marcati
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Sobolev Spaces ◽  
Eigenvalue Problems ◽  
Weighted Sobolev Spaces ◽  
Finite Element Approximation ◽  
Three Dimensions ◽  
Element Approximation ◽  
Singular Potentials ◽  
Dg Method

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

scholarly journals Limit theorems for a diffusion process with a one-sided Brownian potential

2006 ◽  
Vol 43 (4) ◽  
pp. 997-1012
Author(s):  
Kiyoshi Kawazu ◽  
Yuki Suzuki
Keyword(s):  
Asymptotic Behavior ◽  
Diffusion Process ◽  
Limit Theorems ◽  
Random Potential ◽  
Sample Space ◽  
Limiting Behavior ◽  
Maximum Process

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t > 0, the sample space describing the random potential is divided into two parts, Ãt and B̃t, both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ãt or on B̃t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).

scholarly journals A Weighted Eigenvalue Problems Driven by both p(·)-Harmonic and p(·)-Biharmonic Operators

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
Abdelfattah Touzani
Keyword(s):  
Sobolev Spaces ◽  
Eigenvalue Problems ◽  
Biharmonic Operators

AbstractThe existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators\eqalign{& \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left| u \right|^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr}is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp(·)(Ω) and Wm,p(·)(Ω).

Random attractors of FitzHugh–Nagumo systems driven by colored noise on unbounded domains

2019 ◽  
Vol 19 (05) ◽  
pp. 1950035
Author(s):  
Anhui Gu ◽  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
White Noise ◽  
Correlation Time ◽  
Existence And Uniqueness ◽  
Colored Noise ◽  
Stochastic Systems ◽  
Unbounded Domains ◽  
Limiting Behavior ◽  
Uniform Estimates ◽  
Random Attractors

We investigate the pathwise asymptotic behavior of the FitzHugh–Nagumo systems defined on unbounded domains driven by nonlinear colored noise. We prove the existence and uniqueness of tempered pullback random attractors of the systems with polynomial diffusion terms. The pullback asymptotic compactness of solutions is obtained by the uniform estimates on the tails of solutions outside a bounded domain. We also examine the limiting behavior of the FitzHugh–Nagumo systems driven by linear colored noise as the correlation time of the colored noise approaches zero. In this respect, we prove that the solutions and the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise.

Sums and maxima of discrete stationary processes

1993 ◽  
Vol 30 (04) ◽  
pp. 863-876 ◽  
Author(s):  
William P. McCormick ◽  
Jiayang Sun
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Stationary Processes ◽  
Limiting Behavior ◽  
Asymptotic Bounds

This paper considers the joint limiting behavior of sums and maxima of stationary discrete-valued processes. The asymptotic behavior is a cross between a central limit theorem and asymptotic bounds for the distribution of the maxima. Some applications and simulations are also included.

EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

2008 ◽  
Vol 06 (01) ◽  
pp. 83-98 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Positive Real Number ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
Positive Real

We study the boundary value problem - div ((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

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