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Global existence and uniqueness of solutions to a nonlocal phase-field system. (English) Zbl 0938.35035

Bates, P. W. (ed.) et al., Proceedings of the US-Chinese conference: Differential equations and applications, Hangzhou, China, June 24-29, 1996. Cambridge, MA: International Press. 14-21 (1997).
In this paper is proved that, under the assumptions \(j\in C^1 (\mathbb{R})\), \(j(s)=j(-s)\geq 0\) \(\forall s\in\mathbb{R}\), and \(\int_\mathbb{R} j=1\); \(f( \varphi)=\varphi-\varphi^3\); \(\varphi_0\in H^1(\mathbb{R})\) and \(\theta_0\in L^2 (\mathbb{R})\), the following nonlocal phase-field system \[ \varphi_t=j*\varphi-\varphi+ f(\varphi)+\ell\theta, \quad \theta_t+ \ell\varphi_t =\Delta\theta, \quad \varphi(0)= \varphi_0,\;\theta(0) =\theta_0 \] has a unique global solution in an appropriate space.
For the entire collection see [Zbl 0913.00035].

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
76T99 Multiphase and multicomponent flows
45K05 Integro-partial differential equations
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