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Homogenization of attractors for semilinear parabolic equations on manifolds with complicated microstructure. (English) Zbl 0936.35024
The authors investigate a family of parabolic equations: $\partial_t u^\varepsilon- \Delta_\varepsilon u^\varepsilon+ f(u^\varepsilon) =h^\varepsilon(x), \quad x\in M_\varepsilon,\;t>0,$ $\partial_n u^\varepsilon (x)= 0,\;x\in\partial M_\varepsilon,\;t>0,\;u^\varepsilon (x,0)= u^\varepsilon_0 (x),(1)_\varepsilon$ Here $$\varepsilon>0$$ is a small parameter which tends to zero, $$M_\varepsilon\subseteq\mathbb{R}^{n+1}$$ is a certain Riemannian manifold obtained from a fixed domain $$\Omega\subseteq\mathbb{R}^n$$ by a process of homogenization.
The intention of the paper is to show that for every $$\varepsilon>0$$, $$(1)_\varepsilon$$ admits a global attractor $$A_\varepsilon$$ in a suitable functional frame. Moreover there is a limiting equation $$(1)_0$$ of $$(1)_\varepsilon$$ which too admits a global attractor $$A_0$$ such that $$\lim A_\varepsilon=A_0$$ in some sense as $$\varepsilon\to 0$$.
More explicitely let $$\Omega \subseteq\mathbb{R}^n$$ be a smooth bounded domain. A somewhat simplified description of $$M_\varepsilon$$ is as follows. We consider $$\Omega$$ as part of $$\mathbb{R}^{n+1}$$, $$x_{n+1}=0$$. On every lattice point $$\varepsilon j\in\Omega$$, $$j\in\mathbb{Z}^n$$ we put a small sphere $$B_\varepsilon(j) \subseteq\mathbb{R}^{n+1}$$ (with center at $$\varepsilon j)$$ of small radius $$r_\varepsilon$$, all spheres congruent. Let $$\partial B^-_\varepsilon(j)$$ be the part of $$\partial B_\varepsilon(j)$$ in the halfspace $$x_{n+1}\leq 0$$. Now set $M_\varepsilon= \Bigl(\Omega \setminus\bigcup_j B_\varepsilon(j) \Bigr)\cup\bigcup_j \partial B^-_\varepsilon(j).$ On $$M_\varepsilon$$ one can introduce a metric tensor $$g^{\alpha \beta}_\varepsilon (x)$$ which is Euclidean on the part $$M_\varepsilon= \Omega\setminus \bigcup_j B_\varepsilon(j)$$. This metric turns $$M_\varepsilon$$ into a Riemannian manifold which gives rise to the Laplace-Beltrami operator $$\Delta_\varepsilon$$ which occurs in $$(1)_\varepsilon$$. The limiting equations mentioned above split into two parts $\partial_t u-\Delta u+\lambda\mu(u-v)+ f(u)=h_1(x),\;\partial_n u=0\text{ on }\partial \Omega,\;\partial_tv+ \lambda(v-u)+f(v)= h_2(x), (1)_0$ supplied by initial conditions. The precise way in which solutions $$u_\varepsilon$$ of $$(1)_\varepsilon$$ converge toward the solution pair $$u,v$$ of $$(1)_0$$ is somewhat involved; we refer to the paper for details. The same applies to the existence of attractors $$A_\varepsilon,A_0$$ of $$(1)_\varepsilon$$, $$(1)_0$$ respectively and the way in which convergence $$A_\varepsilon\to A_0$$, $$\varepsilon\to 0$$ takes place. Based on this functional frame, the authors proceed to establish a number of theorems. The first, Theorem 3.2, asserts the existence and uniqueness of weak solutions to (2). Theorem 3.2 allows the introduction of the notion of process $$U_f(t,\tau)$$, $$t\geq\tau$$ which describes the evolution of (2). The concepts of absorbing set and attractor are then formulated in terms of $$H(h)$$, $$T^p$$ and $$U_f(t,\tau)$$. The basic theorems now assert the existence of absorbing sets, and based on these, the existence of global attractors.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B41 Attractors 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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