Nonlocal Diffusion Operator Research Articles

Nonlocal dispersal equations with almost periodic dependence. I. Principal spectral theory

This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In this first part of the series, we investigate the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provide various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, establish the relations between them, and study the effect of time and space variations on them. In the second part of the series, we will study the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence applying the principal spectral theory to be developed in this part.

A high order operator splitting method based on spectral deferred correction for the nonlocal viscous Cahn-Hilliard equation

Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method.

Kinetic Solutions for Nonlocal Stochastic Conservation Laws

Abstract This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.

The Dynamics of a Lotka-Volterra competition model with nonlocal diffusion and free boundaries

This paper is concerned with a Lotka-Volterra competition model with nonlocal diffusion and free boundaries, which describes the evolution of an invasive species with free boundaries in a one-dimensional habitat and a native species in the whole space. We prove the well-posedness of the model and obtain a spreading-vanishing dichotomy for the invasive species. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local model. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the local model, where the spreading has finite speed whenever spreading happens.

A local/nonlocal diffusion model

ABSTRACT In this paper, we study some qualitative properties for solutions to an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains. The coupling takes place at the interface between these two domains in such a way that the resulting evolution problem is the gradient flow of an energy functional. We prove existence and uniqueness results, as well as that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Besides, we show a suitable way to recover the heat equation at the whole domain from taking the limit at the nonlocal rescaled kernel. Finally, we propose a brief discussion about the extension of the problem to higher dimensions.

Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

Traveling waves in a nonlocal dispersal SIR model with non-monotone incidence

It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder’s fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator.

The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn–Hilliard equation

AbstractComparing with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal Cahn–Hilliard equations equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. In this paper, we construct an accurate and efficient scalar auxiliary variable approach for the nonlocal Cahn–Hilliard equation with general nonlinear potential. The first contribution is that we have proved the unconditional energy stability for nonlocal Cahn–Hilliard model and its semi‐discrete schemes carefully and rigorously. Second, what we need to focus on is that the nonlocality of the nonlocal diffusion term will lead the stiffness matrix to be almost full matrix which generates huge computational work and memory requirement. For spatial discretizaion by finite difference method, we find that the discretizaition for nonlocal operator will lead to a block‐Toeplitz–Toeplitz‐block matrix by applying four transformation operators. Based on this special structure, we present a fast procedure to reduce the computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy

A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space which has dimension d∈{2,3} and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the finite difference scheme on a uniform mesh. First, we prove the existence and uniqueness of a fully discrete numerical solution of the problem using the Brouwer fixed point theorem. Then, we give a priori bounds and convergence estimates in L2 and L∞ norms for fully-discrete problem. A more delicate analysis in the error provides the second order convergence for the proposed scheme. Numerical results are provided to validate the theoretical analysis.

Coupling local and nonlocal evolution equations

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the decay rates of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media

<p style='text-indent:20px;'>In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions. Next, we obtain some sufficient conditions ensuring the uniform persistence. Finally, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

This paper is mainly concerned with the generalised principal eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised principal eigenvalue. We further investigate the dependence of the generalised principal eigenvalue on the frequency of the periodic oscillation, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency of the periodic oscillation, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.

The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin [2], with its origin tracing back to Capasso et al. [5,6]. We prove that the problem has a unique solution defined for all t>0, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in [2]. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of [2], where the spreading has finite speed whenever spreading happens [33].

Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [Formula: see text] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [Formula: see text]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [Formula: see text] is in a different range. In particular, for the case [Formula: see text], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.

Anisotropic Nonlocal Diffusion Operators for Normal and Anomalous Dynamics

The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit process of isotropic L\'{e}vy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional Laplacian $\Delta_m^{\beta/2}$ and the tempered one $\Delta_m^{\beta/2,\lambda}$ in $\mathbb{R}^n$. Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new nonlocal diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their well-posedness of the Galerkin weak formulation in $\mathbb{R}^n$. To obtain the coercivity, we claim that the probability density function $m(Y)$ should be nondegenerate.

Stability and Error Analysis for a Second-Order Fast Approximation of the Local and Nonlocal Diffusion Equations on the Real Line

The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator, and apply second-order backward differentiation formula (BDF2) to approximate the temporal derivative to achieve a fully discrete infinity system. To solve the resulting fully discrete systems, we develop a unified framework that is applicable to the discretization of both local and nonlocal problems. A key ingredient is to derive Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs). To do so, we apply the $z$-transform and solve an exterior problem using an iteration technique to derive a Dirichlet-to-Dirichlet (DtD)-type mapping as exact ABCs. After that, we use the Green formula to reformulate the DtD-type mapping equivalently as the DtN-type mapping. The resulting DtN-type mapping allows us to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest, and also make it possible to present the stability and convergence analysis of the reduced problem under some open but reasonable assumptions. To efficiently implement the exact ABCs, we further develop a fast convolution algorithm based on approximation of the contour integral induced by the inverse $z$-transform. The stability and error analysis of the reduced finite discrete system based on the fast algorithm for exact ABCs are also established, and numerical examples are provided to demonstrate the effectiveness of our proposed approach.

Existence of solution for a nonlocal dispersal model with nonlocal term via bifurcation theory

In this paper we study the existence of solution for the following class of nonlocal problemsL0u=u(λ−∫ΩQ(x,y)|u(y)|pdy),inΩ, where Ω⊂RN, N≥1, is a smooth bounded domain, p>0, λ is a real parameter, Q:Ω×Ω→R is a nonnegative function, and L0:C(Ω‾)→C(Ω‾) is a nonlocal dispersal operator. The existence of solution is obtained via bifurcation theory.

Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and have received increasing attention in the study of the microstructural evolution of two-phase systems on atomic length and diffusive time scales. This model can be applied to simulate various phenomena such as epitaxial growth, material hardness and phase transition. Compared with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal phase-field crystal equation equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. We propose linear semi-implicit approach and scalar auxiliary variable approach with unconditional energy stability for the nonlocal phase-field crystal equation. The first contribution is that we have proved the unconditional energy stability for nonlocal phase-field crystal model and its semi-discrete schemes carefully and rigorously. Secondly, we found a fast procedure to reduce the computational work and memory requirement which the non-locality of the nonlocal diffusion term generates huge computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small

In this paper, we study the global dynamics of a general $ 2\times 2 $ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper extends the work in [3], where Lotka-Volterra competition models with symmetric nonlocal operators are considered, to more general competition models with nonsymmetric operators.