Asymptotic behavior for nonlocal diffusion equations

Abstract

We study the asymptotic behavior for nonlocal diffusion models of the form u t = J * u − u in the whole R N or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In R N 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., 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.and variations of it, have been recently widely used to model diffusion processes, see [1], [3], [6], [9], [11], [16], [17], [20], [21] and [22]. As stated in [16], if u(x, t) is thought of as the density of a single population at the point x at time t, and J(x−y) is thought of as the probability distribution of jumping from location y to location x, then (J * u)(x, t) = R N J(y −x)u(y, t) dy is

A non-scale-invariant form for coarse-grained diffusion-reaction equations

“…44 However, it can be shown that solutions exist, that the solutions depend continuously on the initial conditions, and that the nonlocal equations converge to the local ones as the nonlocal terms become small enough. [45][46][47][48] Much less information is available about the properties of even linear nonlocal equations in the presence of boundaries, although they are frequently solved numerically. 49 Fortunately, for this particular set of equations, we can make some very compelling arguments indicating that the integral terms are negligible for the purposes of the closure problems.…”

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

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“…The study of semigroup generated by A n−1 has already been done in [1,4] and we only quote their results.…”

Elliptic equations with diffusion parameterized by the range of nonlocal interactions

“…The reader interested with other nonlocal diffusion problems may consult for instance [ZP06] or [CCR06]. This paper is organized as follows.…”