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Asymptotic behavior for nonlocal diffusion equations. (English) Zbl 1126.35081

The authors study the asymptotic behavior for nonlocal diffusion problems of the form \(u_t=J*u-u\), in the whole \(\mathbb R^n\) or in a bounded smooth domain with Dirichlet or Neumann conditions. In the case of \(\mathbb R^n\) the main result is that the long time behavior of 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. For the Dirichlet problem the authors prove that the asymptotic behavior of solutions shows exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile of an eigenfunction of the first eigenvavalue. For the Neumann problem the authors find an exponential convergence to the mean value of the initial condition.

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35B40 Asymptotic behavior of solutions to PDEs
26A33 Fractional derivatives and integrals
35A15 Variational methods applied to PDEs
35R10 Partial functional-differential equations
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