Nonlocal Diffusion Equation Research Articles

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.

Entire Solutions for Nonlocal Dispersal Equations with Bistable Nonlinearity: Asymmetric Case

This paper mainly focuses on the entire solutions of a nonlocal dispersal equation with asymmetric kernel and bistable nonlinearity. Compared with symmetric case, the asymmetry of the dispersal kernel function makes more diverse types of entire solutions since it can affect the sign of the wave speeds and the symmetry of the corresponding nonincreasing and nondecreasing traveling waves. We divide the bistable case into two monostable cases by restricting the range of the variable, and obtain some merging-front entire solutions which behave as the coupling of monostable and bistable waves. Before this, we characterize the classification of the wave speeds so that the entire solutions can be constructed more clearly. Especially, we investigate the influence of the asymmetry of the kernel on the minimal and maximal wave speeds.

Positive solutions for diffusive Logistic equation with refuge

AbstractIn this paper, we study the stationary solutions of the Logistic equation$$\begin{array}{} \displaystyle u_t=\mathcal {D}[u]+\lambda u-[b(x)+\varepsilon]u^p \text{ in }{\it\Omega} \end{array}$$with Dirichlet boundary condition, here 𝓓 is a diffusion operator andε> 0,p> 1. The weight functionb(x) is nonnegative and vanishes in a smooth subdomainΩ0ofΩ. We investigate the asymptotic profiles of positive stationary solutions with the critical valueλwhenεis sufficiently small. We find that the profiles are different between nonlocal and classical diffusion equations.

Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat

This paper is devoted to the study of the propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. We first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super- and sub-solutions. Then we establish the nonexistence, uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument. Finally, we obtain the spreading properties for a large class of solutions.

A difference method for the McKean–Vlasov equation

We analyze a model equation arising in option pricing. This model equation takes the form of a nonlinear, nonlocal diffusion equation. We prove the well posedness of the Cauchy problem for this equation. Furthermore, we introduce a semidiscrete difference scheme and show its rate of convergence.

Spreading speed for a nonlocal diffusive delayed model without quasi-monotonicity

In this article we study the spatial spreading dynamics for a nonlocal diffusive equation with general nonmonotone monostable nonlinearity. On the basis of a sandwich technique and a comparison theorem for solutions of the Cauchy problems of three systems, the spreading speed c⁎ is established for the model. As an application of the results, we consider some models as special cases. It is shown that the results in this article generalize and improve some results in the existing literature.

Nonlocal dispersal cooperative systems: Acceleration propagation among species

We focus on the influences of nonlocal dispersal kernels on the spatial propagation in nonlocal dispersal cooperative systems with initial values having nonempty compact supports. It is well-known that the solution of a monostable nonlocal dispersal scalar equation spreads at a finite speed when the kernel is thin-tailed and propagates by accelerating when the kernel has a heavy tail (the fat-tailed). However, in such systems, we find that one species can propagate by accelerating although its dispersal kernel is thin-tailed, which is a new and interesting phenomenon. More precisely, we show that the spatial propagation of every species at large time is mainly determined by the tail of the maximum of their dispersal kernels, which further implies that their spatial propagation is accelerated by each other because of the cooperation relation between them. This gives us a new understanding of the cooperation relation in the spatial propagation of nonlocal dispersal cooperative systems.

Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media

The present paper is devoted to the study of the existence, the uniqueness and the stability of transition fronts of non-local dispersal equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space non-increasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space non-increasing transition front (if it exists), which implies the uniqueness, up-to-space shifts and monotonicity of transition fronts provided that a space non-increasing transition front exists. Moreover, we show that a transition front must be a periodic travelling front in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space non-increasing transition fronts, whose proof does not need the unbalanced condition.

Final state problem for class of nonlinear nonlocal dispersive equation

We study the large time asymptotics of the solutions to nonlinear dispersive equations of the form(0.1)∂tu−P(−i∂x)∂xu=−∂x(u3),t>0,x∈R, where P(−i∂x) is a pseudo-differential operator defined by the symbol P(ξ). In this paper, we consider the case that P(ξ) is nonlocal and nonhomogeneous and show the existence of the solution to (0.1) around a given modified free solution.

A New Nonlocal Nonlinear Diffusion Equation for Data Analysis

In this paper we introduce and study a new feature-preserving nonlinear anisotropic diffusion for denoising signals. The proposed 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. We provide a mathematical analysis of the existence of the solution of our nonlinear and nonlocal diffusion equation in the two dimensional case (images processing). Finally, we propose a numerical scheme with some numerical experiments which demonstrate the effectiveness of the new method.

Sensitivity filtering from the non-local perspective

We present a mathematical model of topology optimization for non-local linear diffusion equation. The model provides a rigorous and constructive answer to the challenge of interpreting sensitivity filtering as an internal feature of a specific continuum mechanics model, which has been raised in Sigmund and Maute (Struct Multidiscip Optim 46(4):471–475, 2012). Our model enjoys two very interesting properties. On the one hand, the governing non-local integral equations of this model approximate the well-known local generalized Laplace equation with diffusion coefficient obeying SIMP (Solid Isotropic Material with Penalization) law. On the other hand, the topology optimization problem admits optimal solutions for a range of penalization parameters of practical interest without the need for any external regularization techniques, something which is well known to be false in the case of topology optimization with SIMP for classical local description of continuum mechanics.

On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

We consider the Whitham equation ut+2uux+Lux=0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=tanh⁡ξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal C1/2-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol m(ξ), and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol m(ξ) is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.

Non-local Diffusion Equations Involving the Fractional $$p(\cdot )$$-Laplacian

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional $$p(\cdot )$$-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous $$p(\cdot )$$-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem $$({\mathscr {P}}_{1})$$ in a bounded domain $$\varOmega $$ of $${\mathbb {R}}^N$$ and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional $$p(\cdot )$$-Laplacian operator generates a (nonlinear) submarkovian semigroup on $$L^{2}(\varOmega ).$$ In the second part of the paper we establish the existence of global attractors for problem $$({\mathscr {P}}_{2})$$ under certain conditions in the potential $${\mathbb {V}}.$$ Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.

Decay estimates of nonlocal diffusion equations in some particle systems

In this paper, we first consider a nonlocal mixed diffusion equation ∂tu(t,x)=Lu(t,x) in multiparticle systems with different spatial distributions, where the combination of nonlocal diffusion operators L in a spatial variable is defined by the improper integrals with kernels Jj(x) and the low frequencies of Ĵj(ξ) have the same kind of asymptotic expansions. We use the energy method to establish the decay estimates of solutions to ∂tu(t,x)=Lu(t,x) with initial condition u(0, x) = u0(x). Finally, we consider an anisotropic single-particle equation with nonlocal operator L is defined by the improper integrals with kernels Jj(xj) and it is a generation of an anisotropic nonlocal operator −∑j=1N(−∂xjxj)αj.

Weighted [formula omitted] estimates and Fujita exponent for a nonlocal equation

We investigate a nonlocal equation ∂tu=∫RnJ(x−y)u(y,t)dy−‖J‖L1u(x,t)+a(x,t)up in Rn, where a is unbounded and J belongs to a weighted space. Crucial weighted Lp and interpolation estimates for the Green operator are established by a new method based on the sharp Young’s inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ=(1+|x|2∕η)b∕2 that −Γ∕η≲J∗Γ−Γ≲Γ∕η and η−b+∕2≲Γ∕xb≲η−b−∕2. Blow-up behaviors are investigated by employing test functions ϕR=Γ (η=R) instead of principal eigenfunctions. Global well-posedness in weighted Lp spaces for the Cauchy problem is proved. When a∼xσ the Fujita exponent is shown to be 1+(σ+2)∕n. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.

Hölder continuous of the solutions to stochastic nonlocal heat equations

In this paper, we aim to obtain the Hölder continuous of solutions to stochastic nonlocal heat equations. By using Campanato estimates and Sobolev embedding theorem, we first prove the Hölder continuous of the mild solution of stochastic nonlocal diffusion equations locally in Rd in the sense that the solution u belongs to the space Cγ(DT;Lp(Ω)). Then by using tail estimates, we obtain the estimates of the mild solution in Lp(Ω;Cγ(DT)). Lastly, we prove the Hölder continuous of the mild solution on bounded domain.

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

The present paper is devoted to the investigation of the following nonlocal dispersal equationut(t,x)=Dσm[∫ΩJσ(x−y)u(t,y)dy−u(t,x)]+f(t,x,u(t,x)),t>0,x∈Ω‾, where Ω⊂RN is a bounded and connected domain with smooth boundary, m∈[0,2) is the cost parameter, D>0 is the dispersal rate, σ>0 characterizes the dispersal range, Jσ=1σNJ(⋅σ) is the scaled dispersal kernel, and f is a time-periodic nonlinear function of generalized KPP type. This paper is a continuation of the works of Berestycki et al. [3,4], where f was assumed to be time-independent. We first study the principal spectral theory of the linear operator associated to the linearization of the equation at u≡0. We establish an easily verifiable, general and sharp sufficient condition for the existence of the principal eigenvalue as well as important sup-inf characterizations of the principal eigenvalue. Next, we study the influences of the principal spectrum point on the global dynamics and confirm that the principal spectrum point being zero is critical. It is followed by the investigation of the effects of the dispersal rate D and the dispersal range characterized by σ on the principal spectrum point and the positive time-periodic solution. In particular, we prove various limiting properties of the principal spectrum point and the positive time-periodic solution as D,σ→0+ or ∞. To achieve these, we develop new techniques to overcome fundamental difficulties caused by the lack of the usual L2 variational formula for the principal eigenvalue, the lack of the regularizing effects of the semigroup generated by the nonlocal dispersal operator, and the presence of the time-dependence of the nonlinearity f. Finally, we establish the maximum principle for time-periodic nonlocal dispersal operators.

Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay

This paper is concerned with the global stability of non-monotone traveling wave solutions to a nonlocal dispersion equation with time delay. It is proved that, all noncritical traveling wave solutions are globally stable with the exponential convergence rate t−1/αe−μt for some constants μ>0 and α∈(0,2], and the critical traveling wave solutions are globally stable in the algebraic form t−1/α, where the initial perturbations around the monotone/non-monotone traveling wave solution in a weighted Sobolev space can be arbitrarily large. The adopted approach is the anti-weighted energy method combining with Fourier's transform.

Well-posedness for a nonlocal nonlinear diffusion equation and applications to inverse problems

ABSTRACT This article investigates a space–time fractional diffusion equation with a nonlinear source and a space nonlocal operator. The well-posedness is demonstrated from using contraction mapping theorem and generalized Gronwall's inequality firstly. Based on the forward problem, we prove that a nonlinear source term and a fractional order of time derivative are uniquely determined by data which can be observed at one fixed spatial point.

GLOBAL EXISTENCE AND FINITE TIME BLOW-UP OF SOLUTIONS TO A NONLOCAL P-LAPLACE EQUATION

In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].