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63
Asymptotic behavior for nonlocal diffusion equations
 J. Math. Pures Appl
, 2006
"... Abstract. We study the asymptotic behavior for nonlocal diffusion models of the form ut = J ∗ u − u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier t ..."
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Cited by 64 (17 self)
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Abstract. We study the asymptotic behavior for nonlocal diffusion models of the form ut = J ∗ u − u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If Ĵ(ξ) = 1 − Aξα + o(ξα) (0 < α 6 2), the asymptotic behavior is the same as the one for solutions of the evolution given by the α/2 fractional power of the laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition. 1.
HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS
, 2006
"... Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes ..."
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Cited by 40 (17 self)
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Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions. 1.
Boundary fluxes for nonlocal diffusion
"... Abstract. We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the ..."
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Cited by 35 (14 self)
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Abstract. We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition. 1.
A nonlocal pLaplacian evolution equation with nonhomogeneous Dirichlet boundary conditions
 SIAM Journal of Mathematical Analysis
, 2009
"... Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. If p> 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation ut = div(∇up−2∇u) with homogeneous Neuman ..."
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Cited by 33 (12 self)
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Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. If p> 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation ut = div(∇up−2∇u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T;Lp(Ω)) to the solution of the p−laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behaviour of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. 1.
The Neumann problem for nonlocal nonlinear diffusion equations
 J. Evol. Equations
"... Abstract. We study nonlocal diffusion models of the form (γ(u))t(t, x) = Ω J(x − y)(u(t, y) − u(t, x)) dy. Here Ω is a bounded smooth domain and γ is a maximal monotone graph in R2. This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove ex ..."
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Cited by 28 (7 self)
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Abstract. We study nonlocal diffusion models of the form (γ(u))t(t, x) = Ω J(x − y)(u(t, y) − u(t, x)) dy. Here Ω is a bounded smooth domain and γ is a maximal monotone graph in R2. This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove existence and uniqueness of solutions with initial conditions in L1(Ω). Moreover, when γ is a continuous function we find the asymptotic behaviour of the solutions, they converge as t → ∞ to the mean value of the initial condition. 1.
ANALYSIS AND APPROXIMATION OF NONLOCAL DIFFUSION PROBLEMS WITH VOLUME CONSTRAINTS
, 2012
"... Abstract. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between ..."
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Cited by 28 (7 self)
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Abstract. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion of volume constraints are shown to lead to a wellposed problem. Volume constraints are a proxy for boundary conditions that may not be defined for a given kernel. In particular, we demonstrate for a general class of kernels that the nonlocal operator is a mapping between a constrained subspace of a fractional Sobolev subspace and its dual. We also demonstrate for other particular kernels that the inverse of the operator does not smooth but does correspond to diffusion. The impact of our results is that both a continuum analysis and a numerical method for the modeling of anomalous diffusion on bounded domains in Rn are provided. The analytical framework allows us to consider finitedimensional approximations using both discontinuous or continuous Galerkin methods, both of which are conforming for the nonlocal diffusion equation we consider; error and condition number estimates are derived.
A nonlocal convectiondiffusion equation
 J. Functional Analysis
"... Abstract. In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗u−u+G ∗ (f(u)) − f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial c ..."
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Cited by 23 (11 self)
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Abstract. In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗u−u+G ∗ (f(u)) − f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convectiondiffusion equation ut = ∆u+ b · ∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convectiondiffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t → ∞ when f(u) = uq−1u with q> 1. We find the decay rate and the first order term in the asymptotic regime. 1.
A NONLOCAL VECTOR CALCULUS WITH APPLICATION TO NONLOCAL BOUNDARY VALUE PROBLEMS
, 2009
"... Abstract. We develop a calculus for nonlocal operators that mimics Gauss ’ theorem and the Green’s identities of the classical vector calculus for scalar functions. The operators we treat do not involve the gradient of the scalar function. We then apply the nonlocal calculus to define variational fo ..."
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Cited by 21 (8 self)
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Abstract. We develop a calculus for nonlocal operators that mimics Gauss ’ theorem and the Green’s identities of the classical vector calculus for scalar functions. The operators we treat do not involve the gradient of the scalar function. We then apply the nonlocal calculus to define variational formulations of nonlocal “boundary ” value problems that mimic the Dirichlet and Neumann problems for secondorder scalar elliptic partial differential equations. For the nonlocal variational problems, we derive fundamental solutions, show how one can derive existence and uniqueness results, and show how, under appropriate limits, they reduce to their classical analogs.
Decay estimates for nonlocal problems via energy estimates
 Journal de Mathematiques Pures et Applique’es
"... Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x ..."
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Cited by 18 (6 self)
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Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x − y)(u(y, t) − u(x, t)) dy. For example, we will consider equations like, ut(x, t) =
The limit as p→∞ in a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles
"... Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verif ..."
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Cited by 17 (8 self)
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Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator function of the set K = {u ∈ L2(RN) : u(x)−u(y)  ≤ 1, when x − y ∈ supp(J)}. We also find some explicit examples of solutions to the limit equation. If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T;L2(Ω)) to the limit solution of the local evolutions of the p−laplacian, vt = ∆pv. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of MongeKantorovich mass transport theory. 1.