  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.
