## 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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### References:

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