Asymptotic behaviour for a nonlocal diffusion equation on a lattice |
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Authors: | Liviu I Ignat and Julio D Rossi |
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Institution: | (1) Instituto Argentino de Matem?tica—CONICET, Saavedra 15, Piso 3, 1083 Buenos Aires, Argentina |
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Abstract: | In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely,
u¢n(t) = ?j ? \mathbbZd Jn-juj(t)-un(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and
n ? \mathbbZdn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies
?n ? \mathbbZdJn = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform. |
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