Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

TL;DR

This paper develops a new shape index theory for local semiflows in metric spaces, enabling analysis of complex dynamical systems without restrictive assumptions, and applies it to establish solutions for a class of retarded nonautonomous parabolic equations.

Contribution

It introduces the H-shape index theory with generalized shape index pairs, relaxing previous constraints and broadening applicability to non-metrizable and non-separable phase spaces.

Findings

Established the H-shape index theory for local semiflows.

Defined the H-shape cohomology index and Morse equations.

Proved existence of bounded full solutions for a retarded nonautonomous parabolic equation.

Abstract

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions.