Compactly generated shape index theory and its application to a retarded nonautonomous parabolic equation - Wang, Jintao; Li, Desheng; Duan, Jinqiao (Topological Methods in Nonlinear Analysis, 2022)
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 advantagesfor the new shape index theory, in which we obtain a special existence property of bounded full solutions.