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Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term. (English) Zbl 0883.35013
The authors investigate the following problem \[ u_t= \Delta u+(1/\varepsilon^2) f\Biggl(u, \varepsilon\int_\Omega u\Biggr)\quad\text{in }\Omega\times \mathbb{R}^+,\tag{1} \] \[ \partial u/\partial\vec n= 0\quad\text{on }\partial\Omega\times \mathbb{R}^+,\quad u(x,0)= g^\varepsilon(x)\quad\text{for }x\in \Omega,\tag{2} \] where \(\Omega\subset\mathbb{R}^N\) \((N\geq 2)\) is a smooth bounded domain and \(\vec n\) is the unit outward normal to \(\partial\Omega\). Here \(f:\mathbb{R}^2\to \mathbb{R}\) is a smooth function and \(\widetilde f(u)= f(u,0)\) has the properties \[ \widetilde f(\pm 1)= 0,\quad \widetilde f'(\pm 1)<0,\quad \int^1_{-1}\widetilde f(s)ds= 0 \] and there exists a unique value \(a\in(-1, 1)\) such that \(\widetilde f(a)= 0\) with \(\widetilde f'(a)>0\). By the construction of sub-super solutions and by using a comparison principle, the authors prove the main result: There exists a system of continuous functions \(\{g^\varepsilon\}_{0<\varepsilon\leq 1}\) such that the corresponding solutions \(u^\varepsilon\) of problem (1), (2) satisfy the asymptotic condition \(\lim_{\varepsilon\to 0} u^\varepsilon(x, t)=\pm 1\) for any \(x\in\Omega^\pm_t\), \(t\in[0, T^*]\). Here \(\Omega^\pm_t\) are given domains constructed as a geometric motion problem in the first part of this article.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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