Nonlocal System Research Articles

Maximum principles and Liouville results for uniformly elliptic nonlocal Bellman System

In this paper, we are concerned with the uniformly elliptic nonlocal Bellman system. Firstly, we study different maximum principles for the uniformly elliptic nonlocal Bellman operators and obtain key ingredients for carrying on the method of moving planes. Then we derive qualitative properties of solutions to the uniformly elliptic nonlocal Bellman in bounded domains, unbounded domains, and a coercive epigraph domain Ω in Rn, respectively, such as monotonicity, symmetry, and Liouville-type results.

Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity ε2αa+ε4α-3b∫R3|(-Δ)α2u|2dx(-Δ)αu+V(x)u=H(u-β)f(u)inR3,u∈Hα(R3),u>0inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( \\varepsilon ^{2\\alpha }a+\\varepsilon ^{4\\alpha -3}b\\int _{{\\mathbb {R}}^3}|(-\\Delta )^{\\frac{\\alpha }{2}} u|^2{{\\mathrm{d}}}x\\right) (-\\Delta )^\\alpha {u}+V(x)u = H(u-\\beta )f(u) &{} \\quad \ ext{ in }\\,\\,{\\mathbb {R}}^3, \\\\ u\\in H^\\alpha ({\\mathbb {R}}^3),\\quad u>0 &{} \\quad \ ext{ in }\\,\\, {\\mathbb {R}}^3, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon ,beta >0 are small parameters, alpha in (frac{3}{4},1) and a, b are positive constants, (-Delta )^{alpha } is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.

Stability of bistable traveling wavefronts for a nonlocal dispersal epidemic system

This article concerns the stability of traveling wavefronts for a nonlocal dispersal epidemic system. Under a bistable assumption, we first construct a pair of upper-lower solutions and employ the comparison principle to prove that the traveling wavefronts are Lyapunov stable.Then, applying the squeezing technique combining with appropriate upper-lower solutions, we show that the traveling wavefronts are globally exponentially stable. As a corollary, the uniqueness of traveling wavefronts is obtained.

Nonlocal fractional system involving the fractional p, q-Laplacians and singular potentials

In this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities: (S)(-Δ)ps1u=1vα1+vβ1inΩ,(-Δ)qs2u=1uα2+uβ2inΩ,u,v=0in(IRN\\Ω),u,v>0inΩ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (S) \\left\\{ \\begin{array}{ll} (-\\Delta )_{p}^{s_1} u =\\dfrac{1}{v^{\\alpha _1}}+v^{\\beta _1}&{} \ ext { in }\\Omega , \\\\ (-\\Delta )_{q}^{s_2} u =\\dfrac{1}{u^{\\alpha _2}}+u^{\\beta _2}&{} \ ext { in }\\Omega , \\\\ u,v=0 &{} \ ext { in } ({I\\!\\!R}^N\\setminus \\Omega ) , \\\\ u,v>0 &{} \ ext { in } \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega subset {I!!R}^N be a smooth bounded domain, s_1,,s_2in (0,1), alpha _1, alpha _2, beta _1, beta _2 are suitable positive constants, (-Delta )_{p}^{s_1} and (-Delta )_{q}^{s_2} are the fractional p-text {Laplacian} and q-text {Laplacian} operators. Using approximating arguments, Rabinowitz bifurcation Theorem, and fractional Hardy inequality, we are able to show the existence of positive solution to the above system.

On a nonlocal elliptic system of Hardy‐Kirchhoff type with critical exponents

In this work, we consider a class of critical variational systems in of the Hardy‐Kirchhoff type involving the fractional Laplacian operator. By imposing some conditions on the nonlinearity as well as in the potencial, we recover the compactness combining arguments used in Alves and Souto and in Brezis and Nirenberg. Only monotonicity conditions are employed, without imposing any coercivity condition on the potential, which can tend to zero at infinity. Our result is closely related to that obtained by Fiscella, Pucci and Zhang.

Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

We employing a minimization arguments on appropriate Nehari manifolds, we obtain ground state solutionsfor a non-local elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundaryconditions type.

Chaotic self-trapped beams in the nonlocal nonlinear system with the sine-oscillation response

The propagation of the unstable multipeak solitons in the nonlocal nonlinear system with the sine-oscillation response was examined in this paper. It was found that the beams are self-trapped while possessing the chaotic property under both the conditions of negative Kerr coefficient at strong nonlocality and positive Kerr coefficient at weak nonlocality. The self-trapping is represented by the invariant beam width and spectrum width. The chaotic property, denoted by the positive Lyapunov exponents, corresponds to the phenomenon that the profiles of the intensity change irregularly in the propagation direction. Compared with the known chaoticons, which were found only in the strongly nonlocal nonlinear system with the positively defined attenuating response for the positive Kerr coefficient, these beams have different characteristics in the interaction between beams and distribution of energy.

Adaptation to DNA Damage, an Asymptotic Approach for a Cooperative Non-local System

Following previous works about integro-differential equations of parabolic type modelling the Darwinian evolution of a population, we study a two-population system in the cooperative case. First, we provide a theoretical study of the limit of rare mutations and we prove that the limit is described by a constrained Hamilton-Jacobi equation. This equation is given by an eigenvalue of a matrix which accounts for the diffusion parameters and the coefficients of the system. Then, we focus on a particular application: the understanding of a phenomenon called Adaptation to DNA damage. In this framework, we provide several numerical simulations to illustrate our theoretical results and investigate mathematical and biological questions.

On Controllability of Impulsive Fractional Integro-differential Nonlocal System with State

This paper is concerned with the controllability of a nonlinear impulsive fractional integro-differential nonlocal control system with state-dependent delay in a Banach space. At first, we introduce a mild solution for the control system by using fractional calculus and probability density function. Under sufficient conditions, the results are obtained by means of semigroup theory and the Krasnoselskii fixed point theorem. Finally, an example is given to illustrate the main results.

Ground State Solutions for a Nonlocal System in Fractional Orlicz-Sobolev Spaces

We consider an elliptic system driven by the fractional a . -Laplacian operator, with Dirichlet boundary conditions type. By using the Nehari manifold approach, we get a nontrivial ground state solution on fractional Orlicz–Sobolev spaces.

Dynamical analysis of a diffusive malaria model with fixed latent period in the human and vector populations

To investigate the impact of the fixed latent periods in the human and vector populations on the disease transmission in heterogenous environment, we formulate a nonlocal and time-delayed reaction-diffusion (NLTD-RD) system. By appealing to the next generation operator (NGO), we define the basic reproduction number (BRN) [Formula: see text], and prove it as a threshold parameter for indicating whether disease persists or not. Specifically, if [Formula: see text], the disease-free equilibrium is globally asymptotically stable, while if [Formula: see text], the disease is shown to be uniformly persistent. In the homogeneous case that all parameters are assumed to be constants, the explicit expression of [Formula: see text] is obtained. We further achieved the global attractivity of the constant equilibria by utilizing Lyapunov functionals. Numerical simulations are performed to verify the theoretical results and the effects of the diffusion rate on disease transmission.

Weak solution by the sub-supersolution method for a nonlocal system involving Lebesgue generalized spaces

We consider a system of nonlocal elliptic equations $$ \displaylines{ - \mathcal{A}(x, |v|_{L^{r_1(x)}}) \hbox{div}(a_1(|\nabla u|^{p_1(x)})|\nabla u|^{p_1(x)-2}\nabla u)\cr = f_1(x, u,v) |\nabla v|^{\alpha_1(x)}_{L^{q_1(x)}}+g_1(x, u,v) |\nabla v|^{\gamma_1(x)}_{L^{s_1(x)}},\cr - \mathcal{A}(x, |u|_{L^{r_2(x)}}) \hbox{div}(a_2(|\nabla v|^{p_2(x)})|\nabla u|^{p_2(x)-2}\nabla u)\cr = f_2(x, u,v) |\nabla u|^{\alpha_2(x)}_{L^{q_2(x)}}+g_2(x, u,v)|\nabla u|^{\gamma_2(x)}_{L^{s_2(x)}}, } $$with Dirichlet boundary condition, where \(\Omega\) is a bounded domain in\()\mathbb{R}^N\) \((N >1)\) with \(C^2\) boundary. Using sub-supersolution method,we prove the existence of at least one positive weak solution.Also, we study a generalized logistic equation and a sublinear system.

Riemann–Hilbert problems of a nonlocal reverse-time AKNS system of six-order and dynamical behaviors of N-soliton

In this paper, we solve nonlinear nonlocal reverse-time six-component sixth-order AKNS system. We used reverse-time reduction to reduce the coupled system to an integrable sixth-order nonlinear Schrödinger-type equation. Starting from the spectral problem of the AKNS system, a Riemann–Hilbert problem will be formulated. This formulation allows us to generate soliton solutions by using the vectors lying in the kernel of the matrix Jost solutions. When reflection coefficients are zeros, the jump matrix is identity and the corresponding Riemann–Hilbert problem yields soliton solutions, leading to explore their dynamics.

Nonlocal Coupled System for (k,φ)-Hilfer Fractional Differential Equations

In this paper, we study a coupled system consisting of (k,φ)-Hilfer fractional differential equations of the order (1,2], supplemented with nonlocal coupled multi-point boundary conditions. The existence and uniqueness of the results are established via Banach’s contraction mapping principle, the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed-point theorem. Numerical examples are constructed to illustrate the obtained results.

Mittag-Leffler Euler [formula omitted]-differences for Caputo fractional-order systems

Exponential Euler differences have got rapid development recently for integer-order differential equations. But there are few papers focusing on this difference to fractional differential equations. This paper establishes the basic structure of Mittag-Leffler implicit Euler scheme for fractional differential equations by using the constant variation methods in fractional calculus. More critically, since the acquired difference forms belong to the scope of implicit Euler differences and then the fractional PECE algorithms are proposed to solve these implicit differences in numerical calculations effectively. Besides, a novel nonlocal difference operator is proposed and some relative properties are presented. In the end, the existence and boundedness of a unique globally exponentially stable solution of some nonlocal difference system with short-term memory are investigated. Some calculative examples and numerical simulations are given to illustrate the primary research findings.

Foliated Schwarz symmetry of solutions to a cooperative system of equations involving nonlocal operators

In this paper, we prove foliated Schwarz symmetry of solutions to a cooperatively coupled system of equations involving nonlocal operators. Here, the class of nonlocal operators covers in particular the case of the fractional Laplacian. Moreover, we give an explicit example of a nonlocal nonlinear system, in which our result can be applied.

Existence for a Singular Nonlocal Phase Field System with Inertial Term

This paper deals with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $\theta $ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln \theta )_{t}$ , the inertial term and the nonlocal diffusion term for the order parameter $\varphi $ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.2).

The dynamics of traveling wavefronts for a nonlocal delay competition system with local vs. nonlocal diffusions

In this paper, we investigate the existence and exponential stability of traveling wavefronts for a two-component nonlocal delay Lotka–Volterra competition system with local vs. nonlocal diffusions. We first obtain the existence of monostable traveling wavefronts connecting two boundary equilibria by Schauder’s fixed point theorem with an explicit construction of a pair of super- and subsolutions. Furthermore, applying the weighted energy method together with the comparison principle, we prove that all the solutions of the corresponding Cauchy problem converge exponentially to the traveling wavefronts provided that the initial perturbations around the traveling wavefronts belong to a certain weighted Sobolev space.

Stability analysis of a nonlocal SIHRDP epidemic model with memory effects.

The prediction and control of COVID-19 is critical for ending this pandemic. In this paper, a nonlocal SIHRDP (S-susceptible class, I-infective class (infected but not hospitalized), H-hospitalized class, R-recovered class, D-death class and P-isolated class) epidemic model with long memory is proposed to describe the multi-wave peaks for the spread of COVID-19. Based on the basic reproduction number R_0, which is completely controlled by fractional order, the stability of the proposed system is studied. Furthermore, the numerical simulation is conducted to gauge the performance of the proposed model. The results on Hunan, China, reveal that R_0<1 suggests that the disease-free equilibrium point is globally asymptotically stable. Likewise, the situation of the multi-peak case in China is presented, and it is clear that the nonlocal epidemic system has a superior fitting effect than the classical model. Finally an adaptive impulsive vaccination is introduced based on the proposed system. Then employing the real data of France, India, the USA and Argentina, parameters identification and short-term forecasts are carried out to verify the effectiveness of the proposed model in describing the case of multiple peaks. Moreover, the implementation of vaccine control is expected once the hospitalized population exceeds 20% of the total population. Numerical results of France, Indian, the USA and Argentina shed light on the varied effect of vaccine control in different countries. According to the vaccine control imposed on France, no obvious effect is observed even consider reducing human contact. As for India, although there will be a temporary increase in hospitalized admissions after execution of vaccination control, COVID-19 will eventually disappear. Results on the USA have seen most significant effect of vaccine control, the number of hospitalized individuals drops off and the disease is eventually eradicated. In contrast to the USA, vaccine control in Argentina has also been very effective, but COVID-19 cannot be completely eradicated.

Multi-peak solitons in nonlocal nonlinear system with sine-oscillation response

The multi-peak solitons and their stability are investigated for the nonlocal nonlinear system with the sine-oscillation response, including both the cases of positive and negative Kerr coefficients. The Hermite–Gaussian-type multi-peak solitons and the ranges of the degree of nonlocality within which the solitons exist are analytically obtained by the variational approach. This is the first time, to our knowledge at least, to discuss the solution existence range of the multi-peak solitons analytically, although approximately. The variational analytical results are confirmed by the numerical ones. The stability of the multi-peak solitons are addressed by the linear stability analysis. It is found that the upper thresholds of the peak-number of the stable solitons are five and four for the system with negative and positive Kerr coefficients, respectively.