The authors study the long-time behaviour for the Cahn-Hilliard equation, with degenerate mobility and logarithmic singularities. It is given by
\[ \partial_t u=\nabla\cdot(1-u^2)\nabla \bigg(\frac{\Theta}{2}\big(\ln(1+u)-\ln(1-u)\big)-\alpha u-\varepsilon^2\Delta u\bigg), \]
on a bounded domain subject to Neumann-type boundary conditions. This model includes as scaling limits the shallow and deep quench limit.
The long-time behavior of this model is dominated by domains of equilibrium phases \(u_{\pm}\) that coarsen over time, which is characterized by the size of domains where \(u\approx u_{\pm}\). The degree of coarsening is then quantified in terms of a characteristic length scale \(l(t)\) which is prescribed via a Lyapunov functional and a \(W^{1,\infty}\)-norm of the solution.
This paper provides upper bounds on \(l(t)\) for all temperatures \(\Theta>0\) below a critical temperature \(\Theta_c\) and for arbitrary mean concentrations between the pure equilibrium phases. It generalizes the upper bounds obtained by R. V. Kohn and F. Otto [Commun. Math. Phys. 229, No. 3, 375–395 (2002; Zbl 1004.82011)]. In particular, transitions may take place in the nature of the coarsening bounds during coarsening.