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62
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.
Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions
 Israel J. Math
"... Abstract. We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled non local problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation. 1. ..."
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Cited by 16 (2 self)
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Abstract. We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled non local problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation. 1.
The Dirichlet problem for some nonlocal diffusion equations
 Diff. Int. Eq
"... Abstract. We study an adhoc notion of Dirichlet problem for the nonlocal diffusion equation ut = ∫ {u(x + z, t) − u(x, t)} dµ(z), where “u = ϕ on ∂Ω × (0, ∞) ” has to be understood in a nonclassical sense. We then prove existence and uniqueness results of solutions in this setting when µ is a L ..."
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Cited by 15 (5 self)
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Abstract. We study an adhoc notion of Dirichlet problem for the nonlocal diffusion equation ut = ∫ {u(x + z, t) − u(x, t)} dµ(z), where “u = ϕ on ∂Ω × (0, ∞) ” has to be understood in a nonclassical sense. We then prove existence and uniqueness results of solutions in this setting when µ is a L 1 function. Moreover, we prove that our solutions coincide with those obtained through the standard “vanishing viscosity method”, but show that a boundary layer occurs: the solution does not take the boundary data in the classical sense on ∂Ω, a phenomenon related to the nonlocal character of the equation. Finally, we show that in a bounded domain, some regularization may occur, contrary to what happens in the whole space. 1.
Refined asymptotic expansions for nonlocal diffusion equations
"... Abstract. We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole Rd with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form∥∥∥u(u) − ∑ α≤ ..."
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Cited by 14 (4 self)
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Abstract. We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole Rd with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form∥∥∥u(u) − ∑ α≤k (−1)α α! u0(x)xα dx
Traveling wave solutions of spatially periodic nonlocal monostable equations
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Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats
 Proceedings of the American Mathematical Society
"... Abstract. This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equation ..."
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Cited by 11 (6 self)
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Abstract. This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors ’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal. Key words. Monostable equation; nonlocal dispersal; random dispersal; periodic habitat; spreading speed; principal eigenvalue; principal eigenfunction; variational principle.
Asymptotic behavior for a semilinear nonlocal equation, Asympt
 Anal
"... Abstract. We study the semilinear nonlocal equation ut = J∗u− u − up in the whole RN. First, we prove the global wellposedness for initial conditions u(x, 0) = u0(x) ∈ L1(RN) ∩ L∞(RN). Next, we obtain the long time behavior of the solutions. We show that different behaviours are possible dependi ..."
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Cited by 9 (3 self)
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Abstract. We study the semilinear nonlocal equation ut = J∗u− u − up in the whole RN. First, we prove the global wellposedness for initial conditions u(x, 0) = u0(x) ∈ L1(RN) ∩ L∞(RN). Next, we obtain the long time behavior of the solutions. We show that different behaviours are possible depending on the exponent p and the kernel J: finite time extinction for p < 1, faster than exponential decay for the linear case p = 1, a weakly nonlinear behaviour for p large enough and a decay governed by the nonlinear term when p is greater than one but not so large. 1.
A non local inhomogeneous dispersal process
 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 9
"... ABSTRACT. This article in devoted to the the study of the nonlocal dispersal equation ut(x, t) = R ..."
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Cited by 7 (1 self)
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ABSTRACT. This article in devoted to the the study of the nonlocal dispersal equation ut(x, t) = R
Classical, Nonlocal, and Fractional Diffusion Equations on Bounded Domains
 INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING
, 2010
"... The purpose of this paper is to compare the solutions of onedimensional boundary value problems corresponding to classical, fractional and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion, when Fick’s first law is an inaccurate model. ..."
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Cited by 4 (0 self)
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The purpose of this paper is to compare the solutions of onedimensional boundary value problems corresponding to classical, fractional and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion, when Fick’s first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick’s first law in terms of a nonlocal flux is demonstrated to hold. A relationship between nonlocal and fractional diffusion is also reviewed, where the order of the fractional Laplacian can lie in the interval (0, 2]. The contribution of this paper is to present boundary value problems for nonlocal diffusion including a variational formulation that leads to a conforming finite element method using piecewise discontinuous shape functions. The nonlocal Dirichlet and Neumann boundary conditions used represent generalizations of the classical boundary conditions. Several examples are given where the effect of nonlocality is studied. The relationship between nonlocal and fractional diffusion explains that the numerical solution of boundary value problems, where the order of the fractional Laplacian can lie in the interval (0, 2], is possible.
W (2012) Evolution of mixed dispersal in periodic environments
 Discrete and Continuous Dynamical Systems B In
"... Abstract. Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between nonadjacent spatial locations and is thus nonlocal. We propose to study a mixed dispersal strategy, which is a combin ..."
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Cited by 4 (1 self)
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Abstract. Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between nonadjacent spatial locations and is thus nonlocal. We propose to study a mixed dispersal strategy, which is a combination of random dispersal and nonlocal dispersal. More specifically, we assume that a fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes nonlocal dispersal. We investigate how such mixed dispersal affects the invasion of a single species and also how mixed dispersal strategy will evolve in spatially heterogeneous but temporally constant environment. 1. Introduction. Dispersal