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Convergence of a one dimensional Cahn-Hilliard equation with degenerate mobility

We consider a one dimensional periodic forward-backward parabolic equation, regularized by a non-linear fourth order term of order $ε^2\ll 1$. 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 $ε\to0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in $\mathcal{W}^2$ and utilizes the framework of convergence of gradient flows developed by Sandier-Serfaty. As an incidental, we study fine energetic properties of solutions to the Thin-film equation $\partial_tν=(νν_{xxx})_x$.