A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations
The authors investigate the asymptotic behaviour of the scalar parabolic equation \((*)\) \(u_ t = (au_ x)_ x + f(u)\), \(0<x<1\) under boundary conditions \(u_ x(0) = u_ x(1) = 0\). The diffusion coefficient \(a\) is assumed to be smooth, positive and to depend on a small parameter \(\nu>0\) (i.e. \(a=a_ \nu)\) in such a way as to exhibit a certain limiting behaviour as \(\nu \to 0\), which is roughly as follows. It is assumed that there is a fixed point \(x_ 1 \in(0,1)\) with the properties \(\lim_{\nu \to 0} a_ \nu (x_ 1) = 0\) and \(\lim_{\nu \to 0} a_ \nu (x) = + \infty\) for \(x \neq x_ 1\). One of the main parts of the paper consists in the study of the family \(A_ \nu\) of selfadjoint operators, given as follows: \(\text{dom} (A_ \nu) = \{u| u \in H^ 2(0,1)\) and \(u_ x(0) = u_ x (1) = 0\}\), and \(A_ \nu u = - (a_ \nu u_ x)_ x\). Let \(\lambda_ 1 = 0\), \(\lambda_ 2 (\nu) < \lambda_ 3 (\nu) \dots\) be the list of eigenvalues of \(A_ \nu\), and \(\varphi_ 1=1\), \(\varphi_ 2 (\nu)\), \(\varphi_ 3 (\nu), \dots\) a corresponding list of normalized eigenfunctions. The main result about \(A_ \nu\) then states that \(\lim_{\nu \to 0} \lambda_ 2 (\nu) = \zeta\), \(\lim_{\nu \to 0} \lambda_ 3 (\nu) = \infty\) and that \(\lim_{\nu \to 0} \varphi_ 2 (\nu)\) exists and is piecewise constant. Here \(\zeta\) depends on \(x_ 1\) and the data which describe the asymptotic behaviour of \(a_ \nu\). The authors then introduce a two-dimensional ODE: \((**)\;\dot v_ 1 = f_ 1(v_ 1,v_ 2)\), \(\dot v_ 2 = - \zeta v_ 2 + f_ 2 (v_ 1,v_ 2)\). The definition of \(f_ 1,f_ 2\) is in terms of the data which describe the limiting behaviour of \(a_ \nu\). The main result now says, that for any \(\nu\) the system \((*)\) admits a smooth, invariant manifold \(M_ \nu\), which is globally exponentially attracting. The two- dimensional reduced equations on \(M_ \nu\), which describe the flow on \(M_ \nu\), then converge in a specific way toward the ODE \((**)\) as \(\nu \to 0\).
scientific article; zbMATH DE number 635867