On spectral convergence for some parabolic problems with locally large diffusion. (English) Zbl 1419.35105

The authors study the abstract parabolic problem \[ \dot u=-A_{\varepsilon}u+f_{\varepsilon}(u),\tag{1} \] on \(H^1(0,1)\), and a corresponding limit problem \[ \dot z=-A_0z+f_0(z),\tag{2} \] on \(\mathbb{R}^n\), where \((A_{\varepsilon})_{\varepsilon\in (0,\varepsilon_0)}\), (\(\varepsilon_0\in (0,\infty)\)), is a family of linear operators which satisfy some assumptions on a strictly increasing sequence \((x_j)_{j\in [0..n]}\) in \([0,1]\), \(A_0\) is the limit operator for \(\varepsilon\to 0\), and \(f_{\varepsilon}\) and \(f_0\), \(\varepsilon\in (0,\varepsilon_0)\), are Nemitski operators generated by nonlinearities satisfying an appropriate condition. They prove a spectral convergence result and some Conley index continuation principles for the families of local semiflows generated by problems \((1)\) and \((2)\).


35K57 Reaction-diffusion equations
37B30 Index theory for dynamical systems, Morse-Conley indices
35B25 Singular perturbations in context of PDEs
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