## 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 $$(2)$$.

### MSC:

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

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