  This paper investigates a Cahn-Hilliard-Swift-Hohenberg system, focusing on a
three-species chemical mixture subject to physical constraints on volume
fractions. The resulting system leads to complex patterns involving a
separation into phases as typical of the Cahn-Hilliard equation and small scale
stripes and dots as seen in the Swift-Hohenberg equation. We introduce singular
potentials of logarithmic type to enhance the model's accuracy in adhering to
essential physical constraints. The paper establishes the existence and
uniqueness of weak solutions within this extended framework. The insights
gained contribute to a deeper understanding of phase separation in complex
systems, with potential applications in materials science and related fields.
We introduce a stable finite element approximation based on an obstacle
formulation. Subsequent numerical simulations demonstrate that the model allows
for complex structures as seen in pigment patterns of animals and in porous
polymeric materials.
