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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.

MSC:

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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