奇摄动高阶非线性椭圆型方程非局部边值问题

利用微分不等式理论研究了一类奇摄动高阶椭圆型微分方程非局部边值问题．得到了其解一致有效的渐近展开式．     

A logistic type equation in RN with a nonlocal reaction term via bifurcation method

We study the existence of positive solutions of a logistic equation in the entire space with a nonlocal reaction term. Mainly, we apply a bifurcation method and singular boundary equations to obtain a priori bounds of the solutions. Our results show a drastic change of behaviour of the set of positive solutions depending on the sign of the nonlocal term.     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

Positive solution for a class of nonlocal elliptic equations

In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.     

Global Well-Posedness for a Nonlocal Gross–Pitaevskii Equation with Non-Zero Condition at Infinity

We study the Gross–Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with non-zero boundary condition at infinity, in any dimension. We focus on even potentials that are positive definite or positive tempered distributions.     

Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow

We obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow.     

Exact solutions of nonlocal Fokas–Lenells equation

In this paper we propose a nonlocal Fokas–Lenells (FL) equation which can be derived from the Kaup–Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several types of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.     

The obstacle problem for a fractional Monge–Ampère equation

Abstract

We study the obstacle problem for a nonlocal, degenerate elliptic Monge–Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.     

Singular Perturbation in a Nonlocal Diffusion Problem

We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects.

We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.

We study the nonlocal problem both for a single equation and for a system of two equations.

Some of the results obtained are new even when the operator under consideration is the heat operator.     

The well-posedness of a nonlocal hyperbolic model for type-I superconductors

A vectorial nonlocal linear hyperbolic problem with applications in superconductors of type-I is studied. The nonlocal term is represented by a (space) convolution with a singular kernel, which is arising in Eringen's model. The well-posedness of the problem is discussed under low regularity assumptions and the error estimates for two time-discrete schemes (based on backward Euler approximation) are established.     

Local limit of nonlocal traffic models: Convergence results and total variation blow-up

Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.     

A nonlocal multi-point singular fractional integro-differential problem of Lane–Emden type

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane–Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.     

Estimate of the travelling wave speed for an integro-differential equation

Travelling waves for nonlocal reaction–diffusion equations are studied. The minimax representation of the wave speed is obtained. It is used to obtain analytical estimates and asymptotic values of the speed. Two regimes of wave propagation are identified. One of them is dominated by diffusion and another one by the nonlocal interaction.     

Aleksandrov–Bakelman–Pucci maximum principles for a class of uniformly elliptic and parabolic integro-PDE

We prove generalized Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs of Hamilton–Jacobi–Bellman–Isaacs types, whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical Aleksandrov–Bakelman–Pucci maximum principles for the elliptic and parabolic PDEs and an iteration procedure using solutions of Pucci extremal equations. We also provide proofs of nonlocal versions of the classical Aleksandrov–Bakelman–Pucci maximum principles for elliptic and parabolic integro-PDEs.     

An integro-PDE model with variable motility

This paper is concerned with a nonlocal reaction–diffusion–mutation model. It involves the spatial variable and a trait variable which govern the spatial diffusion of species. By establishing comparison principle and constructing monotone iterative sequence, we have proved the existence of solution to Cauchy problem. Then, based on the quasi-elementary solution, auxiliary equation and method of successive improvement of upper and lower solutions, the solutions are shown to be unique, bounded and globally stable.     

Global stability for a nonlocal reaction–diffusion population model

In this paper, we study a nonlocal reaction–diffusion population model. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the model. We then analyze the global stability for the model.     

Positive and negative solutions of a class of nonlocal problems

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem. In this paper, still by using the invariant sets of descent flow, we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows:

     

Evolutions of the momentum density,deformation tensor and the nonlocal term of the Camassa–Holm equation

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.