Nonlocal Diffusion Equation Research Articles

Finite mass solutions for a nonlocal inhomogeneous dispersal equation

In this paper we study the asymptotic behavior of the following nonlocal inhomogeneous dispersal equation$$u_t(x,t) = \int_{\mathbb{R}} J\left(\frac{x-y}{g(y)}\right) \frac{u(y,t)}{g(y)} dy -u(x,t)\qquad x\in \mathbb{R},\ t>0,$$where $J$ is an even, smooth, probability density, and $g$, which accounts for a dispersal distance, is continuous and positive.We prove that if $g(|y|)\sim a |y|$ as $|y|\to + \infty$ for some $0 2$ there are no positivestationary solutions. We also establish the asymptotic behavior of the solutions of the evolution problem in both cases.

Study of visual saliency detection via nonlocal anisotropic diffusion equation

In this paper, a nonlocal anisotropic diffusion equation is constructed to model the evolution of visual saliency. A new multi-directions discretization scheme is adopted to solve the equation. Visual attention diffusion is modeled as a series of diffusion progresses until achieving a stable status. So two stages of diffusion are involved. In the first stage, image boundaries are used to guide the background iterative diffusion process, and the initial saliency seeds are obtained by sorting. In the second stage, an optimization of saliency seeds is executed, and the good saliency seeds are strengthened and updated during each iteration. The final saliency map is obtained through iteratively diffusing saliency scores from the optimal saliency seeds (i.e. the most representative salient elements). Extensive experimental results on two large benchmark databases demonstrate the effectiveness of the proposed method.

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.

Asymptotic behavior of the nonlocal diffusion equation with localized source

Asymptotic behavior of the nonlocal diffusion equation with localized source

Optimal control of a class of nonlocal dispersive equations

In this paper, the optimal control problem for a class of nonlocal dispersive equations is considered. We introduce a kind of the definition of a weak solution to this equation and prove the existence of a unique weak solution in a special Hilbert space S(0,T). We also discuss the existence of optimal control solution to a class of nonlocal dispersive equations with the cost functional. Adopting the Dubovitskii–Milyutin functional analytical approach, we prove the necessary optimality condition of optimal control (local maximum principle) for the investigative system in fixed final horizon case.

A nonlocal dispersal equation arising from a selection–migration model in genetics

This paper is concerned with the existence, uniqueness and asymptotic stability of positive steady-states for a nonlocal dispersal equation arising from selection–migration models in genetics. Due to the lack of compactness and regularity of the nonlocal operators, many classical methods cannot be used directly to the nonlocal dispersal problems. This motivates us to find new techniques. We first establish a criterion on the stability and instability of steady-states. This result is effective to get a necessary condition to guarantee a positive steady-state, it also gives the uniqueness. Then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle. Finally, we consider the dynamic behavior of the initial value problem.

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.

Global stability of travelling wave fronts for non-local diffusion equations with delay

This paper is concerned with the global stability of travelling wave fronts for non-local diffusion equations with delay. We prove that the non-critical travelling wave fronts are globally exponentially stable under perturbations in some exponentially weighted -spaces. Moreover, we obtain the decay rates of using weighted energy estimates.

Minimal wave speed of a nonlocal diffusive epidemic model with temporal delay

This note is concerned with a nonlocal version of the man-environment-man epidemic model in which the dispersion of infectious agents is assumed to follow a nonlocal diffusion law modeled by a convolution operator. The purpose of this note is to show that the minimal wave speeds of properly re-scaled nonlocal diffusion equations can approximate the corresponding one of the classical diffusion equation for this model. As a byproduct, our results indicate that the temporal delay in an epidemic model can reduce the speed of epidemic spread while the nonlocal effect can increase the speed.

Optimal Distributed Control of Nonlocal Steady Diffusion Problems

A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained for the nonlocal diffusion equations and for PDEs. In particular, we observe that using nonlocal models result in a better match to nonsmooth target functions and a reduced cost of control. We also provide a one-dimensional numerical example of nonlocal heat conduction in a bar for which we determine the optimal heat source that results in an optimal match to a reference temperature distribution.

Diffusion-stress coupling in liquid phase during rapid solidification of binary mixtures

Abstract An analytical model has been developed to describe the diffusion-viscous stress coupling in the liquid phase during rapid solidification of binary mixtures. The model starts with a set of evolution equations for diffusion flux and viscous pressure tensor, based on extended irreversible thermodynamics. It has been demonstrated that the diffusion-stress coupling leads to non-Fickian diffusion effects in the liquid phase. With only diffusive dynamics, the model results in the nonlocal diffusion equations of parabolic type, which imply the transition to complete solute trapping only asymptotically at an infinite interface velocity. With the wavelike dynamics, the model leads to the nonlocal diffusion equations of hyperbolic type and describes the transition to complete solute trapping and diffusionless solidification at a finite interface velocity in accordance with experimental data and molecular dynamic simulation.

Weak turbulence plasma induced by two-scale homogenization

This paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the Vlasov–Poisson system. It is shown that the homogenization limit, in the sense of two-scale limit, of the distribution function satisfies the linear Vlasov–Poisson equations. Moreover, the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part (the equilibrium distribution function) is shown to be the solution of the nonlocal quasilinear velocity-space diffusion equation. We also investigate the Landau damping from the point of view of homogenization through the two-scale limit.

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.

Erratum to: Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation

Erratum to: Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation

Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift

We consider some non-local diffusion equations in the presence of a divergence free drift term, where the diffusion operators are related to certain Dirichlet forms of jump type. We derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term. Our class of the velocity includes functions with the scale-critical regularity and some growing functions at spatial infinity.

Notes on symmetric and monotonic solutions of a nonlocal diffusive logistic equation

In this note, using the moving plane method, we establish the properties of symmetric and monotonic solutions of a nonlocal diffusive logistic equation under the assumption of ( f 1 ). Thus, we give a positive answer for the conjecture by Sun, Shi and Wang.

Blow-up phenomena for nonlocal inhomogeneous diffusion problems

This paper is concerned with the blow-up of solutions to some nonlocal inhomogeneous dispersal equations subject to homogeneous Neumann boundary conditions. We establish conditions on nonlinearities sufficient to guarantee that solutions exist for all time as well as blow up at some finite time. Moreover, lower bounds for blow-up time of nonlocal problems are obtained.

Analysis and Comparison of Different Approximations to Nonlocal Diffusion and Linear Peridynamic Equations

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.

Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source , , , , , , and , , which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.