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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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