Compactly generated shape index theory and its application to a retarded nonautonomous parabolic equation
DOI:
https://doi.org/10.12775/TMNA.2021.031Keywords
Local semiflows, compactly generated shape, shape index, Morse equations, retarded nonautonomous equationsAbstract
We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, which allows the phase space to be not separable. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\setminus E$ need not to be a neighbourhood of the compact invariant set. Moreover, we define H-shape cohomology groups and the H-shape cohomology index that is used to develop the Morse equations. Particularly, we apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to illustrate these advantages for the new shape index theory, in which we obtain a special existence property of bounded full solutions.References
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