2013/10/03 by Cecilia Cavaterra, Cavaterra, Cecilia, Maurizio Grasselli +3
Computer Science · Engineering · Materials Science · #35B40 #35B41 #37L99 #80A22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Solidification and crystal growth phenomena #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1310.0965
openalex publication_date 2013/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.