Nonlocal Diffusion Equation Research Articles

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

A numerical method for a nonlocal diffusion equation with additive noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and a Q-Wiener process describing the interactions among the particles in the network and stochastic forcing, respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct semidiscrete (discrete in space and continuous in time) and fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler–Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence is expressed in terms of the regularity of the graphon, the Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows us to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity, including small-world and power-law, are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler–Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints

This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the L∞ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

The High Dimensional Fisher-KPP Nonlocal Diffusion Equation with Free Boundary and Radial Symmetry, Part 1

This is Part 1 of a two-part series, where we study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and obtain a rather complete description of its long-time dynamical behavior, which reveals both similarities and fundamental differences to its one dimensional version considered in [J. Cao, Y. Du, F. Li, and W. T. Li, J. Funct. Anal., 277 (2019), pp. 2772--2814; Y. Du, F. Li, and M. Zhou, J. Math. Pure Appl., 154 (2021), pp. 30--66; Y. Du and W. Ni, Rate of Propagation for the Fisher-KPP Equation with Nonlocal Diffusion, and Free Boundaries, preprint, 2021] recently. Our goals on the long-time dynamics of the model include (a) find the threshold condition on the kernel function that governs the onset of accelerated spreading, (b) determine the spreading speed when it is finite, (c) obtain sharp estimates of the spreading profile for finite speed spreading as well as for accelerated spreading. The tasks in (a) and (b) are carried out here in Part 1, while (c) is done in a separate Part 2. This high dimensional problem poses considerable technical difficulties when the rate of spreading is considered, and we overcome that by introducing an intermediate kernel function which plays a crucial role both in determining the onset condition for accelerated spreading and in obtaining the spreading speed when it is finite.

Mathematical Model for Rice Blast Disease Caused by Spore Dispersion Affected from Climate Factors

Rice blast disease, caused by the fungus Pyricularia oryzae, is one of the diseases that reduce rice yields in Thailand. The fungus’s dispersal influences this disease outbreak. This study aims to develop a mathematical model for spore dispersion, namely a non-local diffusion equation in terms of the dispersal kernel, by adding a factor that causes spore dispersion due to rain splash and adjusting the infection term based on weather conditions such as air temperature and relative humidity. The model assessed the existence and uniqueness of solutions by Banach’s fixed point theorem and used a finite difference method to solve them. The numerical simulation confirmed the existence and uniqueness part of the analysis. Because the climate data was used effectively for developing the disease, the rice blast disease pandemic was widespread in the study areas.

A NEW NONLOCAL NONLINEAR DIFFUSION EQUATION: THE ONE-DIMENSIONAL CASE

AbstractWe prove a result on the existence and uniqueness of the solution of a new feature-preserving nonlinear nonlocal diffusion equation for signal denoising for the one-dimensional case. The partial differential equation is based on a novel diffusivity coefficient that uses a nonlocal automatically detected parameter related to the local bounded variation and the local oscillating pattern of the noisy input signal.

Approximation of random diffusion equations by nonlocal diffusion equations in free boundary problems of one space dimension

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].

Existence and stability of traveling curved fronts for nonlocal dispersal equations with bistable nonlinearity

Existence and stability of traveling curved fronts for nonlocal dispersal equations with bistable nonlinearity

Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces

We study principal eigenvalues and maximum principles for stationary nonlocal operators in spaces of integrable functions defined on general metric measure spaces under minimal assumptions on the kernels. Several characterizations of the principal eigenvalue are given as well as several conditions guaranteeing existence. Characterization of the (strong) maximum principle is also given. For evolution problems we prove the strong maximum principle and characterize stability in terms of the sign of the principal eigenvalue. We recover, extend and improve all previously known results, obtained for smooth open sets in euclidean space under continuity assumptions on the data.

Spreading speeds for the predator-prey system with nonlocal dispersal

In this paper, we mainly study the propagation properties of solutions of the predator-prey system with nonlocal dispersal. More specifically, we explored the spreading speeds of the predator and the prey in two different situations, namely, the predator spreads faster than the prey and the predator spreads slower than the prey. The main difficulty lies in the fact that the comparison principle cannot be used for the predator-prey system. We use the comparison principle of the scalar equation and the method of upper and lower solutions to prove the results. In addition, we establish the comparison principle of nonlocal dispersal equations in space and time dependent environment. We conclude that the predator and the prey will eventually coexist by constructing a suitable Lyapunov functional. Finally, we use numerical simulations to illustrate the results.

On asymptotic behavior for a class of diffusion equations involving the fractional $$\wp (\cdot )$$-Laplacian as $$\wp (\cdot )$$ goes to $$\infty $$

In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $$\wp (\cdot )$$ -Laplacian operator involving constant/variable exponent, with $$\wp ^{-}:=\min _{(x,y) \in {\overline{\Omega }}\times {\overline{\Omega }}} \wp (x,y)\geqslant \max \left\{ 2N/(N+2s),1\right\} $$ and $$s\in (0,1).$$ In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $$\wp $$ goes to $$\infty $$ . Already, for case the weighted fractional $$\wp (\cdot )$$ -Laplacian operator, we will also study the asymptotic behavior of the problem solution when $$\wp (\cdot )$$ goes to $$\infty $$ , in the whole or in a subset of the domain (the problem involving the fractional $$\wp (\cdot )$$ -Laplacian presents a discontinuous exponent). To obtain the results of the asymptotic behavior in both problems it will be via Mosco convergence.

Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form (-Delta )^{beta /2}(log u)le C/t, where beta in (0,2). We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Positive solutions for nonlocal dispersal equation

In the present paper, we consider the positive solutions of a class of degenerate logistic equations with nonlocal dispersal. By employing upper–lower solutions, we establish the existence and uniqueness of positive solutions. Our interesting result exhibits that the spatial degeneracy does not make a change on positive solutions.

Recent developments on spatial propagation for diffusion equations in shifting environments

<p style='text-indent:20px;'>In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented.</p>

The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations

The radial basis function partition of unity (RBF-PU) method  was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently.

Cauchy problem for nonlocal diffusion equations modelling Lévy flights

In the present paper, we study the time-space fractional diffusion equation involving the Caputo differential operator and the fractional Laplacian. This equation describes the Lévy flight with the Brownian motion component and the drift component. First, the asymptotic behavior of the fundamental solution of the fractional diffusion equation is considered. Then, we use the fundamental solution to obtain the representation formula of solutions of the Cauchy problem. In the last, the L 2 -decay estimates for solutions are proved by employing the Fourier analysis technique.

The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in $ {\mathbb R}^3 $

<abstract><p>This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $ N\geq 2 $, in <sup>[<xref ref-type="bibr" rid="b18">18</xref>]</sup>, we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work <sup>[<xref ref-type="bibr" rid="b19">19</xref>]</sup>, we have obtained sharp estimates of the spreading rate when the kernel function $ J(|x|) $ behaves like $ |x|^{-\beta} $ as $ |x|\to\infty $ in $ {\mathbb R}^N $ ($ N\geq 2 $). In this paper, we obtain more accurate estimates for the spreading rate when $ N = 3 $, which employs the fact that the formulas relating the involved kernel functions in the proofs of <sup>[<xref ref-type="bibr" rid="b19">19</xref>]</sup> become particularly simple in dimension $ 3 $.</p></abstract>

Corrigendum to: “Nonlocal dispersal equations with almost periodic dependence. I. Principal spectral theory” [J. Differ. Equ. 295 (2021) 1–38

This note is to fix some gaps in the proofs of some of the results in our paper [M.A. Onyido, W. Shen, J. Differ. Equ. 295 (2021) 1–38].

Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Hölder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L1-stable, which is the first frame work on L1-wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L1-weighted energy estimates combining the compactness analysis, but with some new development.

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.