Nonlocal System Research Articles

A class of non-local elliptic system in non-reflexive fractional Orlicz–Sobolev spaces

The purpose of this paper is to investigate the existence of a weak solution to a non-local elliptic system, driven by the fractional [Formula: see text]-Laplacian operator in fractional Orlicz–Sobolev space that may be non-reflexive. The non-reflexive case occurs when the Orlicz function [Formula: see text] does not verify the [Formula: see text]-condition.

Subordination principle and approximation of fractional resolvents and applications to fractional evolution equations

This paper first analyzes and treats the subordination principle and the approximation of fractional resolvents, extending and perfecting the theory of resolvent. Then, the approximation solvability method and the resolvent technique are developed to investigate the existence of mild solutions to a Riemann-Liouville type fractional nonlocal evolution system without the compactness condition. Moreover, effectiveness of our mentioned findings is supported by an example about a fractional diffusion system.

On the approximate controllability of non-densely defined Sobolev-type nonlocal Hilfer fractional neutral Volterra-Fredholm delay integro-differential system

The main focus of our discussion is the approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral Volterra–Fredholm integro-differential systems. The important outcomes are obtained by applying concepts and ideas from fractional calculus, Dhage’s fixed point theorem, and multivalued maps. Finally, an example is given for deriving theoretical results.

On a nonlocal and nonlinear second-order anisotropic reaction-diffusion system with in-homogeneous Cauchy-Neumann boundary conditions. Applications on epidemic infection spread

<p style='text-indent:20px;'>In our current paper we are following the results obtained by Pavăl et al. in [<xref ref-type="bibr" rid="b36">36</xref>] and study a nonlocal form of the system they propose. First we are performing a qualitative analysis for the equivalent non-local second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction. As in [<xref ref-type="bibr" rid="b36">36</xref>] our PDEs system is also implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. In order to be able to compare with the before mentioned results we use the same hypothesis on the input data: <inline-formula><tex-math id="M1">\begin{document}$ S_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ E_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ I_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ R_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ D_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}</tex-math></inline-formula>, and we prove the well-posedness of a classical solution in <inline-formula><tex-math id="M8">\begin{document}$ C((0,T],C(\Omega)) $\end{document}</tex-math></inline-formula>, extending the types already proven by other authors.</p><p style='text-indent:20px;'>Secondly we construct the implicit-explicit (IMEX) numerical approximation scheme which allows to compute the solution of the system of coupled PDEs. The results are then compared with the ones obtained by [<xref ref-type="bibr" rid="b36">36</xref>].</p>

The influence of immune cells on the existence of virus quasi-species.

This article investigate a nonlocal reaction-diffusion system of equations modeling virus distribution with respect to their genotypes in the interaction with the immune response. This study demonstrates the existence of pulse solutions corresponding to virus quasi-species. The proof is based on the Leray-Schauder method, which relies on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions. Furthermore, linear stability analysis of a spatially homogeneous stationary solution identifies the critical conditions for the emergence of spatial and spatiotemporal structures. Finally, numerical simulations are used to illustrate nonlinear dynamics and pattern formation in the nonlocal model.

Small-signal solutions to a nonlocal cross-diffusion model for interaction of scroungers with rapidly diffusing foragers

As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. For any given conveniently regular and biologically meaningful initial data, smallness conditions on the prescribed resource growth and on the initial nutrient signal concentration are identified which ensure the global existence of a global classical solution to the corresponding no-flux initial-boundary value problem. Moreover, under additional assumptions on the food production source these solutions are shown to be bounded, and to stabilize toward semi-trivial equilibria in the large time limit, respectively.

Gradient continuum model of nonlocal metamaterials with long-range interactions

Compared with classical metamaterials, nonlocal metamaterials have distributed long-range interactions. In this paper, a gradient continuum model is developed to properly predict the dispersive behaviour of a one-dimensional nonlocal metamaterial with long-range interactions. First, a discrete monoatomic model is reconstructed into a supercell model. Then, a Taylor expansion based on supercell model is applied to the continuous displacement field, resulting in a gradient continuum model. The dispersive relation of the gradient continuum model is obtained and compared with discrete supercell model to evaluate its suitability. The proposed gradient continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviours all over the first Brillouin zone. The results indicate that the proposed gradient continuum model can predict the dispersion behaviour of the one-dimensional nonlocal system very well. Furthermore, the gradient continuous model of two mass-in-mass system with long-range interactions are verified.

Interference of Gaussian and/or Airy beams in coupled PT-symmetric nonlocal system

We investigate the interference of Gaussian and/or Airy beams based on coupled parity-time (PT) symmetric nonlocal nonlinear Schrödinger (NNLS) equation, which can be decoupled into two paraxial diffraction equations through a general linear transformation under the balance among nonlocal nonlinear effects. Various superposed solutions of the coupled NNLS equation can be obtained with the help of linear transformation and the solutions of linear diffraction equation. Based on these solutions, we analytically explore the abundant interference phenomena of Gaussian beam train, two Gaussian and/or Airy beams, which can be modulated by the linear transformation, the transverse displacement and relative phase difference between adjacent beams. The results presented here may be helpful to design complex interferometric patterns that can be applied to photoetching or optical printing etc.

A two-dimensional stochastic fractional non-local diffusion lattice model with delays

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

Bi-Dbar-Approach for a Coupled Shifted Nonlocal Dispersionless System

We propose a Bi-Dbar approach and apply it to the extended coupled shifted nonlocal dispersionless system. We introduce the nonlocal reduction to solve the coupled shifted nonlocal dispersionless system. Since no enough constraint conditions can be found to curb the norming contants in the Dbar data, the “solutions” obtained by the Dbar dressing method, in general, do not admit the coupled shifted nonlocal dispersionless system. In the Bi-Dbar approach to the extended coupled shifted nonlocal dispersionless system, the norming constants are free. The constraint conditions on the norming constants are determined by the general nonlocal reduction, and the solutions of the coupled shifted nonlocal dispersionless system are derived.

Existence and multiplicity of the positive normalized solutions to the coupled Hartree–Fock type nonlocal elliptic system

Existence and multiplicity of the positive normalized solutions to the coupled Hartree–Fock type nonlocal elliptic system

Results on approximate controllability for a second-order semilinear nonlocal control system with monotonic nonlinearity

ABSTRACT In this paper, we propose the approximate controllability analysis for a second semilinear system under nonlocal conditions using monotone nonlinearity as a tool. Monotonicity is a significant feature in many communications applications that use digital-to-analog converter circuits. While non-monotonicity prevents such applications from working, nonlinearity makes them work. Therefore, it is of great interest to study a problem while assuming the nonlinear function is monotonic. Nonlocal conditions have a finer effect on the solution and are additional specifications for the physical measurements beyond the conventional ones. In order to determine the controllability of the considered system, we formulate the control function. In addition to this, we have broadened the scope of the results for the impulsive system. At last, we discuss two examples.

Monotonicity of wave profiles for a bistable system on 2-D lattice

This paper is concerned with the monotonicity problem of traveling wave solutions for a class of nonlocal bistable system on two-dimensional lattice. We prove that any traveling wave solution, including the standing wave solution (wave speed c = 0 c=0 ), is strictly monotone increasing for this nonlocal bistable lattice system.

A class of biharmonic nonlocal quasilinear systems consisting of Leray–Lions type operators with Hardy potentials

We study the existence of multiple solutions to a nonlocal system involving fourth order Leray–Lions type operators along with singular terms under Navier boundary conditions. The method is based on the variational methods.

On the initial-boundary value problem for a non-local elliptic-hyperbolic system related to the short pulse equation

In this paper, we prove the well-posedness of the initial-boundary value problem for a non-local elliptic-hyperbolic system related to the short pulse equation. Our arguments are based on energy estimates and passing to the limit in a vanishing viscosity approximation of the problem.

Bifurcation and Pattern Formation in an Activator-Inhibitor Model with Non-local Dispersal.

In this paper, by approximating the non-local spatial dispersal equation by an associated reaction-diffusion system, an activator-inhibitor model with non-local dispersal is transformed into a reaction-diffusion system coupled with one ordinary differential equation. We prove that, to some extent, the non-locality-induced instability of the non-local system can be regarded as diffusion-driven instability of the reaction-diffusion system for sufficiently small perturbation. We study the structure of the spectrum of the corresponding linearized operator, and we use linear stability analysis and steady-state bifurcations to show the existence of non-constant steady states which generates non-homogeneous spatial patterns. As an example of our results, we study the bifurcation and pattern formation of a modified Klausmeier-Gray-Scott model of water-plant interaction.

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.