On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions

In this paper, we investigate the nonlinear fractional difference equation with nonlocal fractional boundary conditions. We derive the Green＇s function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnosel-skii＇s fixed point theorem.     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

非局部边界条件的特征问题及发展问题

考虑在有界区域中非局部边界条件的椭圆特征值问题 ,边界条件与特征值的关系 ,以及在合理假设条件下相应的发展问题的上下解的收敛问题     

具有非局部边界条件的奇摄动反应扩散问题

本文研究了一类具有非局部边界条件的奇摄动反应扩散初始边值问题．在适当的条件下，利用比较定理讨论了问题解的渐近性态．     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

Multiplicity of Solutions to Nonlinear Boundary Value Problem with Nonlocal Boundary Condition

In this paper the author considers the following nonlinear boundary value problem with nonlocal boundary conditions $[\left\{ \begin{array}{l} Lu \equiv - \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ij}}(x)\frac{{\partial u}}{{\partial {x_j}}}) = f(x,u,t)} \u{|_\Gamma } = const, - \int_\Gamma {\sum\limits_{i,j = 1}^n {{a_{ij}}\frac{{\partial u}}{{\partial {x_j}}}\cos (n,{x_i})ds = 0} } \end{array} \right.\]$ Under suitable assumptions on f it is proved that there exists $t_0\in R,-\infinityt_0, at least one solution at t=t_0 at least two solutions as t相似文献   

On a Hele-Shaw Flow Problem with Free and Solid Boundary Components

The purpose of this work is to study a one phase Hele-Shaw fluid flow occupying a time variable domain \(\Omega (t)\), due to the injection of the fluid with a constant rate at a single point of the initial domain \(\Omega (0)\), and in the presence of a fixed solid body \(\Omega _0\). We show the short time existence and uniqueness of the solution for the corresponding boundary value problem in the three dimensional case and in the absence of surface tension.     

Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.     

Dynamics for a Nonlocal Reaction-Diffusion Population Model with a Free Boundary

In this paper we focus on a nonlocal reaction-diffusion population model. Such a model can be used to describe a single species which is diffusing, aggregating, reproducing and competing for space and resources, with the free boundary representing the expanding front. The main objective is to understand the influence of the nonlocal term in the form of an integral convolution on the dynamics of the species. Precisely, when the species successfully spreads into infinity as \(t\rightarrow \infty \), it is proved that the species stabilizes at a positive equilibrium state under rather mild conditions. Furthermore, we obtain a upper bound for the spreading of the expanding front.

     

On a Nonlocal BVP with Nonlinear Boundary Conditions

We consider the existence of at least one positive solution of the problem ${-y''(t)=f(t,y(t)), y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds, y(1)=0}$ , where ${y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds}$ represents a nonlinear, nonlocal boundary condition. We show by imposing some relatively mild structural conditions on f, H ₁, H ₂, and ${\varphi}$ that this problem admits at least one positive solution. Finally, our results generalize and improve existing results, and we give a specific example illustrating these generalizations and improvements.     

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

The solvability of the nonlocal boundary value problem

Inverse Problem with Free Boundary for Heat Equation

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a one-dimensional heat equation with unknown time-dependent leading coefficient in the case where a part of the boundary of the domain is unknown.     

On Free Boundary Problem for the Non-Newtonian Shear Thickening Fluids

The aim of this paper is to explore the free boundary problem for the NonNewtonian shear thickening fluids. These fluids not only have vacuum, but also have strong nonlinear properties. In this paper, a class of approximate solutions is first constructed, and some uniform estimates are obtained for these approximate solutions. Finally, the existence of free boundary problem solutions is proved by these uniform estimates.