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A simple proof of stability of fronts for the Cahn-Hilliard equation. (English) Zbl 1019.35013
The authors prove a theorem (Theorem 1.1) on stability in $L^1$ norm of fronts for the Cahn-Hilliard equation. Consider the Cahn-Hilliard equation on the real line, $$ m_t = (-m_{xx}-1/2 m(1-m^2))_{xx}. \tag 1$$ Let $\overline m_0(x)= \tanh (x/2)$ be a minimizer of the associated free energy $F$, and let $\overline m _a(x)=\overline m(x-a)$ be its translate by $a$. Let $m_0(x)$ be some function that will serve as initial data for (1), let $m(t)$ be the solution of (1) through $m_0(x)$, and define $a$ by $\int_{\Bbb R } (m_0(x)-\overline m_a(x)) dx = 0$. Suppose that $\int_{\Bbb R }x^2(m_0(x)- \overline m_0(x))^2 dx \leq c_0$ for some positive constant $c_0$. Then for any $\varepsilon > 0$ there is a positive constant $\delta=\delta(\varepsilon, c_0)$ such that if $\int_{ \Bbb R }(m_0(x)-\overline m_0(x))^2 dx \leq \delta$, we have that $$ \|m(t)- \overline m_a \|_1 \leq c_2(1+c_1 t)^{-(5/52-\varepsilon)}, $$ where $a$ is as defined above and $c_1$, $c_2$ depend on $\varepsilon$ and $c_0$. Similar algebraic decay estimates hold for the excess free energy. The main ingredients in the dissipation-dichotomy argument are the lower bound on the dissipation rate (Lemma 1.4), the moment inequality of Theorem 1.5, the constrained uncertainty principle (1.24) and the consequences of the differential inequalities (1.30), that elegantly (since $9/13>1/2$) allow the authors to derive the $L^1$ decay estimates. The paper is not self-contained and uses freely results from previous work by the authors [J. Phys. 95, 1069-1117 (1999; Zbl 0931.35174) and Commun. Partial Differ. Equations 25, 847-886 (2000; Zbl 0954.35030)]. The missing reference on p. 339 is to (1.29).

MSC:
35B35Stability of solutions of PDE
35K30Higher order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
35K55Nonlinear parabolic equations
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