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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)

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
Full Text: DOI
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