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Non-autonomous lattice systems with switching effects and delayed recovery. (English) Zbl 1345.34128

The authors study the following non-autonomous lattice dynamical system with a discontinuous nonlinear term, and a special type of recoverable delay \[ \begin{aligned} &\dot u_i=\nu(u_{i-1}-2u_i+u_{i+1})+f_i(t,u_i)\cdot {\mathcal \chi}[L_i-\max_{-\theta\leq s\leq 0}u_i(t+s)], \quad i\in \mathbb Z, \quad t>t_0, \\ &u_i(\tau)=\phi_i(\tau-t_0), \quad \forall \tau\in [t_0-\theta, t_0], \quad i\in \mathbb Z, \quad t_0\in \mathbb R, \end{aligned} \] where \(\nu =1/R>0\) is the coupling coefficient, \(R\) is the intercellular resistance, \({\mathcal \chi}\) is the Heaviside function, and for each \(i\in \mathbb Z u_i\in \mathbb R\) represents the membrane potential of the cell at the \(i\)-th active site, \(L_i\in \mathbb R\) is the threshold triggering, the switch-off at the \(i\)-th site, \(u_i(t+\cdot)\in C([-\theta,0],\mathbb R)\) is the segment of \(u_i\) on time interval \([t-\theta,t]\) where \(\theta\) is a positive constant, and \(f_i \in C(\mathbb R\times\mathbb R, \mathbb R)\) satisfies proper dissipative conditions.
The problem is formulated as an evolution inclusion with delays and the existence of weak and strong solutions is established. The long term behavior is investigated and the existence of a pullback attractor is proved.

MSC:

34K31 Lattice functional-differential equations
34K05 General theory of functional-differential equations
34K09 Functional-differential inclusions
34K30 Functional-differential equations in abstract spaces
92C37 Cell biology
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