Asymptotic behavior for nonlocal diffusion equations

By Emmanuel Chasseigne, M. Chaves, Julio D. Rossi. 2006

341 citations recorded by the source corpus.

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 Jˆ(ξ)=1−A|ξ|α+o(|ξ|α) (0<α⩽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 yet 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.

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Citations in the corpus

Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (2014, background): “2) are often used to model diffusive systems which exhibit nonlocal internal interactions (see [4, 7, 10, 11, 12, 16, 19, 24, 26], etc.”
Representation of solutions and large-time behavior for fully nonlocal diffusion equations (2015, background): “Our emphasis is on the decay properties, and for the space-fractional heat diffusion such questions have been studied, for instance, by Chasseigne, Chaves and Rossi in [13] as well as by Ignat and Rossi in [26].”
Nonlocal problems with Neumann boundary conditions (2014, background): “The idea of [4, 8] (and also [9, 10, 11]) is to consider the regional fractional Laplacian, associated to the Dirichlet form cn,s ∫”
Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats (2013, unknown): “k(z) > 0 if ‖z‖ < r0 and k(z) = 0 if ‖z‖ ≥ r0 for some r0 > 0, where ‖ · ‖ denotes the norm in R ), and ∫ RN k(z)dz = 1 (see [1], [7], [13], [18], [22], etc.”
On the principal eigenvalue of some nonlocal diffusion problems (2009, background): “The particular instance of the parabolic problem in RN when f = 0 is considered in [9], [27], while the “Neumann” boundary condition for the same problem is treated in [1], [16] and [17].”
A nonlocal convection–diffusion equation (2007, background): “This is due to the fact that the nonlinear convection is of higher order and that the radially symmetric diffusion leads to gaussian kernels in the asymptotic regime, see [5] and [15].”
Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “See, for example, [2], [6], [10], [12], [14], [19], [20], [25], [26], [30], [31], [32], [34], [53], for the study of spectral theory for nonlocal dispersal operators and the existence, uniqueness, and stability of nontrivial positive stationary solutions.”
A logistic equation with refuge and nonlocal diffusion (2009, background): “The particular instance of the parabolic problem in RN when f = 0 is considered in [7], [27], while”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, unknown): “The reader is referred to [5,8,10,11,13,15,16,19,20,22,24–26,28,32], etc.”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, unknown): “The reader is referred to [1], [2], [4], [5], [6], [7], [8], [10], [11], [16], [22], [23], [24], [26], [28], [30], [32], etc.”
Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats (2012, background): “See, for example, [2], [7], [8], [10], [12], [15], [16], [19], [22], [23], [24], [26], [40], [41], for the study of spectral theory for nonlocal dispersal operators and the existence, uniqueness, and stability of nontrivial positive stationary solutions.”