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

Abstract

In this work, we study the Dirichlet problem for a class of non-linear coupled systems of reaction–diffusion non-local type:  

\[ \begin{cases} u_t-a_1(l_1(u),l_2(v))\Delta u+\lambda_1|u|^{p-2}u=f_1&\hbox{in }\Omega\times ]0,T],\\ v_t-a_2(l_1(u),l_2(v))\Delta v+\lambda_2|v|^{p-2}v=f_2&\hbox{in } \Omega\times ]0,T],\\ u=v=0&\hbox{on } \partial\Omega\times ]0,T],\\ u=u_0,\ v=v_0&\hbox{in }\Omega\times\{0\}. \end{cases} \]

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 in Chipot & Lovat (1997, Some remarks on non-local elliptic and parabolic problems. Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), vol. 30. pp. 4619–4627), Corrêa et al. (2004, On a class of problems involving a nonlocal operator. Appl. Math. Comput., 147, 475–489), Raposo et al. (2008, Solution and asymptotic behaviour for a non-local coupled system of reaction–diffusion. Acta Appl. Math., 102, 37–56) and in Simsen & Ferreira (2014, A global attractor for a nonlocal parabolic problem. Nonlinear Stud., 21, 405–416) for coupled systems.