We consider that the stellar initial mass function (IMF) depends on physical properties of star-forming molecular clouds in galaxies and thereby investigate how such a non-universal IMF (NUIMF) influences galaxy evolution. We incorporate a NUIMF model into galaxy-scale chemodynamical simulations in order to investigate the differences in chemical and dynamical evolution of disk galaxies between the NUIMF and universal IMF (UIMF) models. In the adopted NUIMF model, the three slopes of the Kroupa IMF depend independently on densities and metallicities ([Fe/H]) of molecular gas clouds, and production rates of metals and dust from massive and AGB stars can vary according to the time evolution of the three IMF slopes. The preliminary results of the simulations are as follows. Star formation rates in actively star-forming disk galaxies can be significantly lower in the NUIMF model than in the UIMF model, and the differences between the two models can be more remarkable in galaxies with higher SFRs. Chemical enrichment can proceed more rapidly in the NUIMF model and [Mg/Fe] for a given metallicity is higher in the NUIMF model. The evolution of H2 fraction (f_H2) and dust-to-gas ratio (D) is more rapid in the NUIMF model so that the final f_H2 and $D$ can be higher in the NUIMF model. Formation of massive stellar clumps in gas-rich disks is more strongly suppressed owing to the stronger SN feedback effect in the NUIMF model. The radial density profiles of new stars within the central 1kpc are shallower in the NUIMF model. ; Comment: 27 pages, 19 figures, accepted for the publication in MNRAS