Existence and continuity of uniform exponential attractors of the singularity perturbed Hodgkin-Huxley system
It is not possible in this short review to discuss satisfactorily the authors paper which comprises seventy pages. We thus have to content us with a few indications. The author studies rate of attraction and fractal dimension of two systems, both intimately connected with the Hodgkin-Huxley system. The first of these is \[ \left. \begin{aligned} u_t &-\Delta u+ f_1(w)u= f_2(w))\\ w_t &+ h_1(u)w= h_2(u)\end{aligned} \right\}\quad\text{in }\Omega,\tag{1} \] \[ \partial_nu= 0\quad \text{on }\partial\Omega, \] where \(\Omega\subseteq \mathbb{R}^N\) is a smooth bounded domain and \(\partial_n u(P)\) the derivative with respect to the outer unit normal at \(P\in\partial\Omega\). The second system reads as follows: \[ \left. \begin{aligned} \varepsilon u_{tt} &+ (\varepsilon f_3(w)+ 1)u_t- \Delta u+ f_1(w)u= f_3(w)\\ w_t &+ h_1(u)w= h_2(u)\end{aligned} \right\}\quad\text{in }\Omega,\tag{2} \] \[ \partial_nu= 0\quad\text{on }\partial\Omega. \] The functions \(f_i\), \(h_j\) together with their derivatives are assumed to be continuous and uniformly bounded and such that (1) admits a positive invariant region. Systems (1), (2) give rise to semiflows, namely \[ \begin{aligned} S_\varepsilon(t)(u_0, v_0, w_0) &= (u^\varepsilon(t), u^\varepsilon_t(t), w^\varepsilon(t)),\\ S(t)(u_0, w_0) & = (u(t), w(t)).\end{aligned} \] Here \(u^\varepsilon(t)\), \(u^\varepsilon_t(t)\), \(w^\varepsilon(t)\) is the solution to (2) for initial conditions \(u_0\), \(u_t(0)= v_0\), \(w_0\); likewise with \(S(t)\). Both \(S(t)\) and \(S_\varepsilon(t)\) act on suitable function spaces. In order to express the main result one has to replace \(S_\varepsilon(t)\) by an extension \(S^\varepsilon(t)\) which acts on four-vectors: \[ S^\varepsilon(t)(u_0, v_0,\sqrt\varepsilon v_0,w_0)= (u^\varepsilon(t), u^\varepsilon_t(t), \sqrt\varepsilon u^\varepsilon_t(t), w^\varepsilon(t)). \] The principal result now states that for \(\varepsilon\) sufficiently small, \(^\varepsilon(t)\), \(t\geq 0\) has an exponential attactor \(M_\varepsilon\) of \(E_0\)-\(E_1\)-type in a suitable space of four-vectors \((u,u_t,\sqrt\varepsilon u_t,w)\). The fractional dimension of \(M_\varepsilon\) is uniform with respect to \(\varepsilon\). The notion ``of type \(E_0\)-\(E_1\)'' is of technical nature and refers to the differential spaces involved. The rate of attraction of trajectories toward \(M_\varepsilon\) is uniform in \(\varepsilon\). The main novelty in this statement is in the independence and uniformity in \(\varepsilon\) of fractional dimension and rate of convergence. There are further statements of a similar kind which refer in part to (1) and \(S(t)\), and to (2) and \(S^\varepsilon(t)\) but with a different pair of function spaces \(E_0\)-\(E_1'\) involved. The remaining fifty-five pages are devoted to the proof which cannot be discussed here.
scientific article; zbMATH DE number 1206647