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We establish the compactly generated shape (H-shape) index theory for localsemiflows on complete metric spaces via more general shape index pairs, anddefine the H-shape cohomology index to develop the Morse equations. The mainadvantages are that the quotient space $N/E$ is not necessarily metrizable forthe shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of thecompact invariant set. Moreover, in this new theory, the phase space is notrequired to be separable. We apply H-shape index theory to an abstract retardednonautonomous parabolic equation to obtain the existence of bounded fullsolutions.

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