A compact dissipative dynamical system for a difference equation with diffusion (Q1365212)

A discrete-time version of the parabolic scalar equation is investigated. Consider a sequence \(\{u_n\}_{n\in\mathbb{N}}\) of functions \(u_n\in H^1_0=\{p:[0,1]\to\mathbb{R}: p\) is absolutely continuous with \(p(0)=p(1)=0\) and \(p'\in L_2(0,1)\}\) satisfying: \[ u_{n+1}-u_n=\varepsilon\Delta u_{n+1}+\varepsilon f\circ u_n,\quad n\in\mathbb{N},\;\Delta=\partial^2/\partial x^2,\text{ for given }\varepsilon>0. \tag{1} \] The aim of the paper is to study the iteration properties of the map \(\Phi_\varepsilon:H^1_0\to H^1_0\), \(\Phi_\varepsilon=(I-\varepsilon\Delta)^{-1}\circ(I+\varepsilon\hat f)\), where \(\hat f(p)=f\circ p\) for \(p\in H^1_0\) and \(I\) is the identity of \(H^1_0\) . Remark that the sequence \(\{u_n\}_{n\in\mathbb{N}}\) satisfies (1) if and only if \(u_{n+1}=\Phi_\varepsilon(u_n)\), \(n\in\mathbb{N}\).