| Summary: | We consider a one-dimensional periodic forward-backward parabolic equation, regularized by a degenerate nonlinear fourth-order term of order $\varepsilon^2\ll 1$. This equation is known in the literature as Cahn--Hilliard equation with degenerate mobility. Under the hypothesis of the initial data being well prepared, we prove that as $\varepsilon\to0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in $\mathcal{W}^2(\Bbb{T})$ and utilizes the framework of convergence of gradient flows developed by Sandier and Serfaty (Comm. Pure Appl. Math., 57 (2004), pp. 1627--1672; Discrete Contin. Dyn. Syst., 31 (2011), pp. 1427--1451). As an incidental, we study fine energetic properties of solutions to the thin-film equation $\partial_t\nu=-(\nu\nu_{xxx})_x$.
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