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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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