Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations. (English) Zbl 1334.37015

Summary: We consider the following lattice nonclassical diffusion equation with delays \[ \dot{v}_i(t) + \lambda_0 v_i(t) +(- 1)^p \Delta^p v_i(t) + \varepsilon(- 1)^p \Delta^p \dot{v}_i(t) = f_i(v_i(t - \rho(t))) + g_i(t),\quad i \in \mathbb{Z}, \] where \(\varepsilon \in(0, 1], \lambda_0\) is a positive constant with \(\lambda_0 < 1, p\) is any positive integer and \(\Delta\) is the discrete one-dimensional Laplace operator. Under suitable conditions on \(f\) and \(g\) we prove the existence of pullback attractors for the multi-valued process associated with the \(\varepsilon\)-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the \(\varepsilon\)-small perturbed systems, and show that their attractors are “close” in the sense of Hausdorff semidistance.


37B55 Topological dynamics of nonautonomous systems
34A33 Ordinary lattice differential equations
35B41 Attractors
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
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