A reaction–diffusion model for the non-local coupled system: existence, uniqueness, long-time behaviour and localization properties of solutions

In this work, we study the Dirichlet problem for a class of nonlinear coupled systems of reaction-diffusion nonlocal type:  ut − a1(l1(u), l2(v))∆u+ λ1|u|u = f1(x, t) in Ω×]0, T ] vt − a2(l1(u), l2(v))∆v + λ2|v|v = f2(x, t) in Ω×]0, T ] u(x, t) = v(x, t) = 0 on ∂Ω×]0, T ] u(x, 0) = u0(x), v(x, 0) = v0(x) in Ω . Email addresses: jduque@ubi.pt (José C.M. Duque), ralmeida@ubi.pt (Rui M.P. Almeida ), anton@ptmat.fc.ul.pt (Stanislav N. Antontsev ), ferreirajorge2012@gmail.com (Jorge Ferreira ) URL: http://www.cmatubi.ubi.pt (José C.M. Duque), http://cmaf.ptmat.fc.ul.pt (Stanislav N. Antontsev ) Preprint submitted to Elsevier July 6, 2013 We prove the existence and uniqueness of weak and strong solutions of these systems and localization properties of the solutions, including the waiting time effect. Moreover important results on polynomial and exponential decay and vanishing of the solutions in finite time are also presented. We improve the results obtained by Chipot and Lovart [1], Corrêa, Menezes and Ferreira [2], Raposo et al. [3] and Simsen and Ferreira [4] for coupled systems.

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