On spectral convergence for some parabolic problems with locally large diffusion (Q1741790)

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 $(2)$.