Nonlocal Equation Research Articles

Pointwise estimates on the fundamental solution to nonlocal diffusion equation

In this article we study the pointwise behavior of the fundamental solution S ( t , x ) to the nonlocal diffusion model u t = D ( J ∗ u − u ) on R , S ( t , x ) can be written as e − Dt δ ( x ) + S 0 ( t , x ) , where S 0 ( t , x ) is called the regularizing part. Under some conditions on J and its Fourier transform J ^ , we obtain some new pointwise estimates on S 0 ( t , x ) . Then we apply our results to study the spatial dynamics of the nonlocal diffusion population model in a shifting environment considered in Li et al. [Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J Nonlinear Sci. 2018;28:1189–1219]. In our study the convolution kernel J can be assumed to be more general, i.e. not necessarily of compact support, also the proof is greatly simplified. It is our believe that these pointwise estimates possess their own interest in further studying nonlocal diffusion models.

Rogue waves in a reverse space nonlocal nonlinear Schrödinger equation

Rogue waves in a reverse space nonlocal nonlinear Schrödinger (NLS) equation with real and parity-symmetric nonlinearity-induced potential are considered. This equation has clear physical meanings since it can be derived from the Manakov system with a special reduction. The N-fold Darboux transformation and its generalized form for the nonlocal NLS equation are constructed. As an application, the multiparametric Nth-order rogue wave solution in terms of Schur polynomials for the nonlocal NLS equation with focusing case is derived by the limit technique. The significant differences of rogue wave dynamics between the nonlocal NLS equation and its usual (local) counterpart are illustrated through two types of specific rogue wave solutions. Unlike the eye-shaped (Peregrine type) rogue waves, the rogue wave doublets which involve an eye-shaped rogue wave and a dark/four-petaled rogue wave merging or separating with each other, and the rogue wave sextets that are characterized by the superpositions of three eye-shaped rogue waves and three dark/four-petaled rogue waves with fundamental, triangular and quadrilateral patterns are shown. Moreover, some wave characteristics including the difference between the light intensity and the plane-wave background, and the pulse energy of the rogue wave doublets are discussed.

Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.

On positive solutions of the Cauchy problem for doubly nonlocal equations

We study the Cauchy problem for the doubly nonlocal equation where and for some m ∈ ℝ and 0 < α ≤ +∞, here α = +∞ means that J is compactly supported. The problem not only describes nonlocal diffusions and nonlocal interactions, but also it is a nonlocal counterpart of the nonlocal heat equation in [10] where and m > 0. As for the local existence of positive solutions, we find that problem (0.1) is quite different from the nonlocal heat equation. That is, problem (0.1) admits positive solutions if and only if m > n − q(n + α), however the nonlocal heat equation has positive solutions for any m ∈ ℝ. For m ≥ 0 or n(1 − q) < m < 0 with α = +∞, we determine the critical Fujita curve of problem (0.1) as F(p, q, n, m, r, α) := (p − 1)[nq − (n − m)+] − q(β − nr) = 0 with β = min{α, 2}. That is, if F ≤ 0, every positive solution of (0.1) blows up in finite time, whereas if F > 0, there exist both nonglobal solutions and global solutions. In the supercritical case F > 0, we further discuss the second critical exponent which describes the critical decay rate of initial data to distinguish both solutions. Particularly, we find a new phenomenon, namely, if with α = +∞, the second critical exponent is independent of p and r.

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Positive solutions for a nonhomogeneous nonlocal logistic equation

In this paper, we deal with the nonhomogeneous nonlocal dispersal equation∫ΩJ(x−y)u(y)dy−u(x)+λu(x)−a(x)up(x)+Mf(x)=0,x∈Ω¯, where Ω is a bounded and smooth domain, p>1 and the parameters M,λ>0. The kernel function J(x) is nonnegative and symmetric, and the coefficients a,f∈C(Ω¯) are assumed to be nonnegative. We are interested in the existence, uniqueness and asymptotic profiles of positive solutions. It is shown that the structure of positive solutions makes a fundamental change due to the nonhomogeneous effect. Moreover, we study the sharp profiles of positive solutions when there is a spatial degeneracy of a(x). The result exhibits that the blow-up profiles are determined by the degeneracy of a(x).

Global regularity for critical SQG in bounded domains

AbstractWe prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

Tempered nonlocal integrodifferential equations with nonsmooth solution data

This work on the time discretization for the solution of the tempered nonlocal integro-differential equation with smooth and nonsmooth initial data are extended. We find the difficulty arising from theoretical analysis can be overcome by Laplace transform technique. A mount of proof skills have been provided based on Laplace transform technique for this kind of equations. The Laplace transform method is employed effectively to show that the proposed interpolating quadrature scheme is convergence for smooth and non-smooth initial data in both homogeneous and inhomogeneous cases.

Cordes-Nirenberg type results for nonlocal equations with deforming kernels

We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Ampère equation. Under a natural integrability assumption on the Monge-Ampère solution, we prove a stability lemma allowing the ellipticity class to vary. Using a compactness method, we then derive Hölder regularity estimates for the gradient of the solutions.

Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains

Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains

Analysis of Non-Local Integro-Differential Equations with Hadamard Fractional Derivatives: Existence, Uniqueness, and Stability in the Context of RLC Models

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam’s stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer’s fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding.

Non-local time evolution equation with singular integral and its application to traffic flow model

We consider an integro-differential equation model for traffic flow, which extends the Burgers equation model. To derive exact solutions to the equation, we first examine the features of integrable integro-differential equations and find that they are obtained through the residue formula from integrable equations in complex domains. Using this approach, we construct several new integrable equations with double singular integrals and elliptic singular integrals. Then, we discuss the traffic model and show that it exhibits interactions between free-flow and congested regions, as well as the deadlock phenomenon, depending on the non-locality parameter.

Approximate Controllability of a Coupled Nonlocal Partial Functional Integro-differential Equations with Impulsive Effects

Approximate Controllability of a Coupled Nonlocal Partial Functional Integro-differential Equations with Impulsive Effects

Nonlocal Elliptic and Parabolic Equations with General Stable Operators in Weighted Sobolev Spaces

Nonlocal Elliptic and Parabolic Equations with General Stable Operators in Weighted Sobolev Spaces

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

Mechanism for the formation of an inhomogeneous nanorelief and bifurcations in a nonlocal erosion equation

Продолжены исследования нелокального уравнения эрозии, которое используется в качестве математической модели формирования пространственно неоднородного рельефа на поверхности полупроводников. Показано, что формирование такого рельефа может происходить в результате локальных бифуркаций при смене устойчивости у пространственно однородного состояния равновесия. Рассмотрена периодическая краевая задача, у которой изучаются бифуркации коразмерности 2. Для решений, описывающих неоднородный нанорельеф, получены асимптотические формулы, и изучен вопрос об их устойчивости. Анализ математической задачи основан на современных методах теории динамических систем с бесконечномерным фазовым пространством, в частности на методе интегральных многообразий и аппарате теории нормальных форм.

Riemann–Hilbert approaches of an M-coupled nonlinear Schrödinger system with variable coefficients and the associated nonlocal equation

Riemann–Hilbert approaches of an M-coupled nonlinear Schrödinger system with variable coefficients and the associated nonlocal equation

The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable method, the highly efficient and accurate schemes (in time) with unconditional energy stability for solving the nonlocal Cahn-Hilliard equation are proposed. On the one hand, we have demonstrated the unconditional energy stability and mass conservation for the nonlocal Cahn-Hilliard equation with its high-order numerical schemes in the continuous and discrete level carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on fast Fourier transform and conjugate gradient approach for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.