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Convective nonlocal Cahn-Hilliard equations with reaction terms. (English) Zbl 1335.35096

This paper is devoted to a nonlocal Cahn-Hilliard-type initial-boundary value problem of the form
\[ \begin{cases} \dfrac{\partial\varphi}{\partial t}+\nabla\cdot(u\varphi) + K(x, \varphi)=\Delta\mu + g &\text{in } \Omega\times(0,T),\\ \displaystyle\mu=\varepsilon (J\star 1) \varphi-\varepsilon J\star\varphi+\frac{1}{\varepsilon} f(\varphi) &\text{in } \Omega\times(0,T),\\ \displaystyle\frac{\partial\mu}{\partial n}=0 &\text{on }\partial\Omega\times (0,T), \\ \displaystyle \varphi(0)=\varphi_0 &\text{in } \Omega, \end{cases}\tag{1} \]
where \(\Omega\) is a regular bounded domain by \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\) and \(N \leq 3\). The authors are interested in two different forms of \(K\). In the first scenario, \( K(x, \varphi) = \sigma (\varphi - \varphi^{*})\), \(\sigma > 0\) and \(\varphi^{*} \in \mathbb{R}\), in which case (1) models, for instance, pattern formation in diblock-copolymers as well as binary alloys with induced reaction and type-I superconductors. In the second scenario, \( K(x, \varphi) = \lambda_0 \chi_{\Omega\backslash D}(x) (\varphi - h),\, \lambda_0 > 0\); in that case, (1) has applications in binary image inpainting (\(h\) is the given damaged binary image). The term \(\nabla\cdot(u\varphi)\) represents a transport term which accounts for a possible flow of the mixture at a certain given velocity field \(u\) and \(g\) is a given external source.
Conditions are derived on the nonlinearities and on the kernel convolution under which the global existence of a unique solution is obtained in both cases. Furthermore, they prove the existence of global attractors of the dynamical systems associated with the problems.

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
35B41 Attractors
35K41 Higher-order parabolic systems
45K05 Integro-partial differential equations
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References:

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