Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 55, Issue 5, Pages 5823-5861Publisher
SIAM PUBLICATIONS
DOI: 10.1137/22M1540429
Keywords
stabilization of solutions; gradient-type systems; convective Cahn-Hilliard-type equation
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This paper shows the stabilization of solutions to the sixth-order convective Cahn-Hilliard equation. By applying an abstract result by Carvalho, Langa, and Robinson, it is proved that for small delta, the equation has the structure of gradient flow in a weak sense. The desired stabilization is obtained through a powerful theorem due to Hale and Raugel. The importance of this article lies in providing a significant approach to solve the problem by citing other research and applying theorems.
We show stabilization of solutions to the sixth-order convective Cahn-Hilliard equation. The problem has the structure of a gradient flow perturbed by a quadratic destabilizing term with coefficient delta > 0. Through application of an abstract result by Carvalho, Langa, and Robinson we show that for small delta the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilization follows from a powerful theorem due to Hale and Raugel.
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