Convergence to Equilibrium for the Cahn-Hilliard Equation with a Logarithmic Free Energy
Helmut Abels and Mathias Wilke
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Submission date: 28. Jun. 2006
published in: Nonlinear analysis / A, 67 (2007) 11, p. 3176-3193
DOI number (of the published article): 10.1016/j.na.2006.10.002
MSC-Numbers: 35K55, 35B40, 35Q99, 47H05, 47J35, 80A22
Keywords and phrases: Cahn-Hilliard equation, logarithmic potential, Lojasiewicz-Simon inequality, convergence to steady states, monotone operators
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In this paper we investigate the asymptotic behavior of the nonlinear Cahn-Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz-Simon inequality to show that each solution converges to a steady state as time tends to infinity.