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This work investigates the well-posedness and optimal control of asixth-order Cahn-Hilliard equation, a higher-order variant of the celebratedand well-established Cahn-Hilliard equation. The equation is endowed with asource term, where the control variable enters as a distributed mass regulator.The inclusion of additional spatial derivatives in the sixth-order formulationenables the model to capture curvature effects, leading to a more accuratedepiction of isothermal phase separation dynamics in complex materials systems.We provide a well-posedness result for the aforementioned system when thecorresponding nonlinearity of double-well shape is regular and then analyze acorresponding optimal control problem. For the latter, existence of optimalcontrols is established, and the first-order necessary optimality conditionsare characterized via a suitable variational inequality. These results aim atcontributing to improve the understanding of the mathematical properties andcontrol aspects of the sixth-order Cahn-Hilliard equation, offering potentialapplications in the design and optimization of materials with tailoredmicrostructures and properties.

Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation