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An $$L^ \infty$$ bound for solutions of the Cahn-Hilliard equation. (English) Zbl 0851.35010
This paper deals with the initial value problem in $$x\in \mathbb{R}^n$$, $$t> 0$$: $\partial_t u^\varepsilon= \Delta u^\varepsilon+ \sum^n_{i, j= 1} \partial_{ij} f_{ij}(u^\varepsilon, x)- \varepsilon^2 \Delta^2 u^\varepsilon,\;u^\varepsilon(x, 0)= u^\varepsilon_0(x),$ where $$\varepsilon> 0$$ is a small parameter, $$u^\varepsilon$$ is real-valued, and $$f_{ij}(\lambda, x)$$, $$i, j= 1,\dots, n$$ are constants for $$|\lambda |\geq 1$$, uniformly in $$x\in \mathbb{R}^n$$. Under suitable assumptions on the regularity of the data, the authors show that $$u^\varepsilon$$ is uniformly bounded in $$\mathbb{R}^n\times (0, + \infty)$$, $$\varepsilon> 0$$. The proof is mostly based on properties of the convolution kernel $$N(x, t)= \overline{\mathcal F}[e^{- (|\xi|^2+ |\xi|^4)t}]$$, where $$\overline{\mathcal F}$$ denotes the inverse Fourier transform.
Some corollaries and generalizations are mentioned.
Reviewer: D.Huet (Nancy)

##### MSC:
 35B25 Singular perturbations in context of PDEs 35K30 Initial value problems for higher-order parabolic equations 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
##### Keywords:
Cahn-Hilliard equation
Full Text:
##### References:
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