  We explore the effect of inhomogeneity on electronic properties of the
two-dimensional Hubbard model on a square lattice using dynamical mean-field
theory (DMFT). The inhomogeneity is introduced via modulated lattice hopping
such that in the extreme inhomogeneous limit the resulting geometry is a Lieb
lattice, which exhibits a flat-band dispersion. The crossover can be observed
in the uniform sublattice magnetization which is zero in the homogeneous case
and increases with the inhomogeneity. Studying the spatially resolved
frequency-dependent local self-energy, we find a crossover from Fermi-liquid to
non-Fermi-liquid behavior happening at a moderate value of the inhomogeneity.
This emergence of a non-Fermi liquid is concomitant of a quasi-flat band. For
finite doping the system with small inhomogeneity displays $d$-wave
superconductivity coexisting with incommensurate spin-density order, inferred
from the presence of oscillatory DMFT solutions. The $d$-wave superconductivity
gets suppressed for moderate to large inhomogeneity for any finite doping while
the incommensurate spin-density order still exists.
