Narrow Results

We study $s$-fractional $p$-Laplacian-type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+\sigma ,q}$ estimates for any choice of pairs $(\sigma ,q)$ with $q\in (p,\infty )$ and $\sigma \in (0,min\{\frac{s}{p-1},1-s\})$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a nonhomogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.

In this paper, we investigate a nonlocal multi-point and multi-strip coupled boundary value problem of nonlinear fractional Langevin equations. The standard fixed point theorems (Leray–Schauder’s alternative and Banach’s fixed point theorem) are applied to derive the existence and uniqueness results for the given problem. We also discuss the Ulam–Hyers stability for the given system. Examples illustrating the obtained results are presented. Some new results appearing as special cases of the present ones are also indicated.

Inspired by some recent works of Lovelock Brans–Dicke (BD) gravity and mimetic gravity, cosmology solutions in extensions of these two modified gravities are investigated. A nonlocal term is added to the Lovelock BD action and Gauss–Bonnet (GB) terms to the mimetic action, correspondingly. de Sitter and power scale factor solutions are then obtained in both theories. They can provide natural new approaches to a more accurate description of the unverse evolution.

A generalization of the two-dimensional Yang–Mills and generalized Yang–Mills theory is introduced in which the building B-F theory is nonlocal in the auxiliary field. The classical and quantum properties of this nonlocal generalization are investigated and it is shown that for large gauge groups, there exists a simple correspondence between the properties of the nonlocal theory and its corresponding local theory.

We study the phase structure of nonlocal two-dimensional generalized Yang–Mills theories (nlgYM₂) and it is shown that all order of ϕ^2k model of these theories has phase transition only on compact manifold with g = 0 (on sphere), and the order of phase transition is 3. Also it is shown that the model of nlgYM₂ has third order phase transition on any compact manifold with , and has no phase transition on the sphere.

Modeling networks of synaptically coupled neurons often leads to systems of integro-differential equations. Particularly interesting solutions in this context are traveling waves. We prove here that spectral stability of traveling waves implies their nonlinear stability in appropriate function spaces, and compare several recent Evans-function constructions that are useful tools when analyzing spectral stability.

This paper is concerned with the global dynamics of a Lotka–Volterra competition diffusion system having nonlocal intraspecies terms. Based on the reconstructed comparison principle and monotone iteration, the existence and uniqueness of the solution for the corresponding Cauchy problem are established. In addition, the spreading speed of the system with compactly supported initial data is considered, which admits uniform upper and lower bounds. Finally, some sufficient conditions for guaranteeing the existence and nonexistence of Turing bifurcation are given, which depend on the intensity of nonlocality. Comparing with the classical Lotka–Volterra competition diffusion system, our results indicate that a nonconstant periodic solution may exist if the nonlocality is strong enough, which are also illustrated numerically.

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L^∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration.

Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases.

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators ${({\partial }_t-\mathcal{ℒ})}^s$, $0<s<1$, where $\mathcal{ℒ}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators ${\left(-\mathcal{ℒ}\right)}^s$. Our focus is on non-Euclidean situations.

This paper presents the snap-through and bifurcation elastic stability analysis of nano-arch type structures with the Winkler foundation under transverse loadings by the strain gradient and stress gradient (nonlocal) theories. The equations of equilibrium are derived by using the variational method and virtual displacement theorem of minimum total potential energy. In the elastic stability analysis, von Karman's nonlinear strain component is included, with the deformation represented by a series solution. It is concluded that in general, the strain gradient theory pushes the system away from instability as compared to the classical theory. However, the nonlocal theory does the reverse and causes the system to experience instability earlier than that of the classical theory. Moreover, theories with different small-size considerations change the mechanism of instability in different ways. For example, in similar conditions, the strain gradient theory causes the system to reach a snap-through point, while the nonlocal theory causes the system to stop at a bifurcation critical point.

Instabilities in nanosized, externally pressurized spherical shells are important for their applications in nano and biotechnology. Mechanics at such length scale is described by nonlocal and Strain Gradient (SG) field theories. However, analysis of shell buckling is involved and becomes even more complicated in presence of nonlocal and SG interactions. This paper demonstrates that such analysis can be largely simplified by a shallow segment representation of the shell by assuming short wave lengths for the incipient buckling modes. The governing equations are derived and linearized equations are solved to obtain a closed form solution for the critical external pressure causing buckling for a pressurized nonlocal shell. Nonlocal interactions are shown to reduce, whereas the SG interaction increases the critical pressure. The relative reduction/increase becomes more prominent for higher modes of buckling and for increasingly thinner shell. A constricting relationship between the two set of wave numbers expressing the buckling modes is also shown to be modified by the nonlocal and SG scale parameters. Consequent wave numbers increase/decrease, accompanied by decreasing/increasing number of wavelengths, thereby further justifying the shallow segment representation employed herein.

The aim of this work is to investigate the free vibration and buckling characteristics of sigmoid functionally graded (FG) nanoplate with the influence of porosity. The modified rule of the mixture is utilized to calculate the effective material properties of porous sigmoid functionally graded (P-SFGM) nanoplate. Three schemes of porosity distribution, including uniform, symmetric and nonsymmetric are investigated. The first-order shear deformation theory is utilized to simulate the displacement fields of P-SFGM nanoplate. Eringen’s nonlocal elastic theory and isogeometric analysis (IGA) are used to establish the governing equations for free vibration and buckling analysis of nanoplate structure with small size effect. By using NURBS as a basic function, IGA can fulfill the higher-order derivative requirement of governing equations. The accuracy of the presented solution is verified. By taking the nonlocal parameter into account, the stiffness of the plate is softened. Also, the effects of porosity distribution across the plate’s thickness, porosity parameter, material power index, boundary conditions (BCs) and aspect ratio on the frequency response of P-SFGM nanoplate are presented.

We study uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities. We give also conditions which lead to a lack of minimizers and we show how minimization on an infinite dimensional space reduces here to a minimization on ℝ. Among other things, we prove that uniqueness of minimizers of functionals of the form ∫_Ω a(∫_Ω gu dx)|∇u|² dx - 2 ∫_Ω fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a)″ < 0 on (0, ∞).

This paper deals with propagation of Rayleigh-type waves in nonlocal inhomogeneous transversely isotropic half-space. From the characteristic equation of wave for the nonlocal inhomogeneous transversely isotropic medium, the existence of the number of surface waves depends on the inhomogeneity of the medium through the number of solutions satisfying the damping condition of the characteristic equation discussed. It has also been concluded that there exist cut-off frequency and escape frequency for the wave propagating in size-dependent materials based on the nonlocal theory. Dispersion equation for the propagation of Rayleigh-type surface waves at the free surface has been derived. Based on the obtained dispersion equation, the effects of the inhomogeneity of material and nonlocality parameter on the Rayleigh wave propagation are considered through some numerical examples.

In this paper, we present a finite element scheme with Crank–Nicolson method for solving nonlocal parabolic problems involving the Dirichlet energy. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of dual-phase lag (DPL) theory of thermoelasticity based on Eringen’s nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for temperature, displacement and stress within the rod is carried out and displayed graphically. The effect of moving heat source speed, time instance, memory-dependent derivative, magnetic-field and nonlocality on temperature, displacement and stress are studied.

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups $SO(32)$ and $E_8\times E_8$. The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin-$\frac{1}{2}$ fields transform under the adjoint representation in these models, compactification of the heterotic string over $G_2/SU(3)$ or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over $\frac{G_2\times SU(2)\times U(1)}{SU(3)\times U{(1)}^{\prime }\times U{(1)}^{\prime \prime }}$, provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin-$\frac{1}{2}$ fields now would transform under the fundamental representation of $G_2\times SU(2)\times U(1)$ and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the $G_2\times SU(2)\times U(1)$ model vanishes at $n_{\frac{1}{2}}\simeq 103.34271924$. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions.

The presence of size effects represented by a small nanoscale on torsional wave propagation properties of circular nanostructure, such as nanoshafts, nanorods and nanotubes, is investigated. Based on the nonlocal elasticity theory, the dynamic equation of motion for the structure is formulated. By using the derived equation, simple analytical solutions for the relation between wavenumber and frequency via the differential nonlocal constitutive relation and the numerical solutions for a discrete nonlocal model via the integral nonlocal constitutive relation have been obtained. This results not only show that the dispersion characteristics of circular nanostructures are greatly affected by the small nanoscale and the classical theory overestimates the stiffness of nanostructures, but also highlights the significance of the integral nonlocal model which is able to capture some boundary characteristics that do not appear in the differential nonlocal model.

In this paper, we are dedicated to studying the global dynamics of a nonlocal predator–prey model with double mutation. First, by defining a pair of upper and lower solutions, we build a new comparison principle. Furthermore, based on the new comparison principle, we get the existence of the solutions by constructing monotone iterative sequences. Finally, using the quasi-fundamental solution, Gronwall’s inequality and auxiliary functions, the uniqueness and uniform boundedness of the solutions are given. It is worth noting that this paper is the first to introduce the double mutation into the system and obtain the well-posedness and the uniform boundedness of the solutions by more detailed analysis and estimation.