Homogeneous Boundary Research Articles

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation: $$ - \varepsilon \Delta _p u = f(x,u)in\Omega ,$$ where 1 0 is a small parameter, $$f(x,u) = \left\{ \begin{gathered} \left| u \right|^{p - 2} u\left| {u - a(x)} \right|^{p - 2} (u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),ifp \geqslant 2 \hfill \\ \left| u \right|^{(2 - p)/(p - 1)} u(u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),if1 < p < 2, \hfill \\ \end{gathered} \right. $$ where ω > 0, a(x) is a continuous function satisfying 0 < a(x) < 1 for x ∈ \(\bar \Omega \), Ω is a bounded smooth domain in ℝN. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.

Blow-up for an evolution p-laplace system with nonlocal sources and inner absorptions

Abstract This paper investigates the blow-up properties of positive solutions to the following system of evolution p-Laplace equations with nonlocal sources and inner absorptions { u t − div ( | ∇ u | p − 2 ∇ u ) = ∫ Ω v m d x − α u r , x ∈ Ω , t &gt; 0, v t − div ( | ∇ v | q − 2 ∇ v ) = ∫ Ω u n d x − β v s , x ∈ Ω , t &gt; 0 with homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω ∈ R N (N ≥ 1), where p, q &gt; 2, m, n, r, s ≥ 1, α, β &gt; 0. Under appropriate hypotheses, the authors discuss the global existence and blow-up of positive weak solutions by using a comparison principle. 2010 Mathematics Subject Classification: 35B35; 35K60; 35K65; 35K57.

Rate-independent processes with linear growth energies and time-dependent boundary conditions

A rate-independent evolution problem is considered for which the stored energy density depends on the gradient of the displacement. The stored energy density does not have to be quasiconvex and is assumed to exhibit linear growth at infinity; no further assumptions are made on the behaviour at infinity. We analyse an evolutionary process with positively $1$-homogeneous dissipation and time-dependent Dirichlet boundary conditions.

Matrix-based Chebyshev spectral approach to dynamic analysis of non-uniform Timoshenko beams

A Chebyshev spectral method (CSM) for the dynamic analysis of non-uniform Timoshenko beams under various boundary conditions and concentrated masses at their ends is proposed. The matrix-based Chebyshev spectral approach was used to construct the spectral differentiation matrix of the governing differential operator and its boundary conditions. A matrix condensation approach is crucially presented to impose boundary conditions involving the homogeneous Cauchy conditions and boundary conditions containing eigenvalues. By taking advantage of the standard powerful algorithms for solving matrix eigenvalue and generalized eigenvalue problems that are embodied in the MATLAB commands, chebfun and eigs, the modal parameters of non-uniform Timoshenko beams under various boundary conditions can be obtained from the eigensolutions of the corresponding linear differential operators. Some numerical examples are presented to compare the results herein with those obtained elsewhere, and to illustrate the accuracy and effectiveness of this method.

Modeling Hazardous Gas Dispersion in Neutrally and Stably Stratified Atmospheric Boundary Layer Flows

The objective of the present work is to propose a methodology to simulate hazardous gas dispersion in neutrally and stably stratified atmospheric boundary layer (ABL) flows. The profiles of velocity, temperature and turbulence quantities are set based on the stratification effect. Special emphasis is placed on the generation of a horizontally homogeneous turbulent boundary layer (HHTBL). FLUENT software is validated for simulating ABL flows by comparing the results obtained with available numerical data from the literature. The fully developed profiles of velocity, turbulent kinetic energy, turbulent dissipation rate and temperature are implemented as user defined functions in the commercial code FLUENT. The hazardous gas release is modeled using the convection diffusion equation. Four test cases from the classical Prairie Grass experiment are considered for validating the present approach. The predicted concentration of the SO2 gas at various locations of the test field (50, 100, 200, 400 and 800 m) are ...

Modified SPR Technique for 3D Tensor-Product Block Finite Elements: A Computer-Based Test

In this paper, the Poisson problem with homogeneous Dirichlet boundary conditions on the bounded open domain is considered and the superconvergence property of 3D tensor-product block finite elements is analyzed by using a computer-based test.

Steady states for resource and sexual competition models with diffusions

We study the existence and nonexistence of positive solutions of resource and sexual competition models under a homogeneous Dirichlet boundary condition. In addition, when one of the coefficients of sexual competition is absent, the results for the multiplicity and stability of positive solutions are obtained. The methods employed are the upper-lower solution technique and the fixed point theory in a positive cone.

Blow-up solutions for localized reaction-diffusion equations with variable exponents

Communicated by W. Sprosig This paper deals with radial solutions to localized reaction-diffusion equations with variable exponents, subject to homogeneous Dirichlet boundary conditions. The global existence versus blow-up criteria are studied in terms of the variable exponents. We proposed that the maximums of variable exponents are the key clue to determine blow-up classifications and describe blow-up rates for positive solutions. Copyright © 2011 John Wiley & Sons, Ltd.

Mixed Fourier-Jacobi Spectral Method for Two-Dimensional Neumann Boundary Value Problems

AbstractIn this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.

Qualitative analysis of a strongly coupled prey-predator model

This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non-existence of the non-constant positive solutions are obtained. Moreover, we prove that the nonlinear diffusion terms can create non-constant positive equilibrium solutions when the corresponding model without nonlinear diffusion term fails.

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

Free Vibrations of Beams with Arbitrary Boundary Conditions Based on Wave Propagation Method

Under the Euler-Bernoulli beam theory, the wave propagation method is used for the vibration analysis of beams with arbitrary boundary conditions. The boundary conditions end the beam could be arbitrary that all the conventional homogeneous beam boundary conditions can be included by setting the stiffnesses of the springs be infinity or zero. In this paper, the flexural displacement of the beam is expressed in the wave propagation form including wave numbers. The wavenumber could be obtained in a known form for conventional boundary conditions. So the results are obtained through the boundary conditions and the known wavenumbers and compared with the numerical results. In order to validate the correctness, results with different stiffness are compared with those obtained by previous published papers.

Parametric optimization for a hyperbolic equation in divergence form with a pointwise state constraint: II

On the basis of the results of the first part of the paper, we consider necessary conditions for minimizing sequences in an optimal control problem with a pointwise state constraint of inequality type and with dynamics described by a linear hyperbolic equation in divergence form with the homogeneous Dirichlet boundary condition. The state constraint contains a function parameter that belongs to the class of continuous functions and occurs as an additive term. For the parametric optimization problem, we also consider regularity and normality conditions stipulated by the differential properties of its value function.

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.

Non-Constant Positive Steady States for a Predator-Prey Cross-Diffusion Model with Beddington-DeAngelis Functional Response

This paper deals with a predator-prey model with Beddington-DeAngelis functional response under homogeneous Neumann boundary conditions. We mainly discuss the following three problems: (1) stability of the nonnegative constant steady states for the reaction-diffusion system; (2) the existence of Turing patterns; (3) the existence of stationary patterns created by cross-diffusion.

Weak Solution to a Parabolic Nonlinear System Arising in Biological Dynamic in the Soil

We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction‐diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.

Initial-Boundary Value Problem for the Camassa–Holm Equation with Linearizable Boundary Condition

We present a Riemann–Hilbert problem formalism for the initial boundary value problem for the Camassa–Holm equation on the half-line x > 0 with homogeneous Dirichlet boundary condition at x = 0. We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann–Hilbert problem determined by the initial data via associated spectral functions. This allows us to apply the non-linear steepest descent method and to describe the large-time asymptotics of the solution.

Gibbs phenomenon and its removal for a class of orthogonal expansions

We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u t = f(u)(Δu + a∫Ω u(x, t)dx − u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.

Hölder-Type Approximation for the Spatial Source Term of a Backward Heat Equation

We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation with a homogeneous Cauchy boundary condition, where ϕ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term f and provide Hölder-type error estimates in both L 2 and H 1 norms. Numerical experiments are provided.