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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.

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
Full Text: DOI
[1] Babin, A. V.; Vishik, M. J., Attractors of Evolution Equations (1992), Amsterdam: North-Holland, Amsterdam
[2] L.Boutet de Monvel - E.Ya. Khruslov,Homogenization on Riemannian manifolds, BiBoS Preprint 560/93, Bielefeld (1993) (to appear in the Proceedings «Composite Media», Trieste, 1994). · Zbl 0909.35016
[3] Chueshov, I. D., A problem on non-linear oscillations of shallow shell in a quasistatic formulation, Math. Notes, 47, 401-407 (1990) · Zbl 0721.73028
[4] Chueshov, I. D., Global attractors for non-linear problems of mathematical physics, Russian Math. Surveys, 43, 3, 133-161 (1993) · Zbl 0805.58042
[5] Chueshov, I. D., Quasistatic version of the system of von Karman equations, Matem. Physika, Analiz, Geometriya, 1, 1, 149-167 (1994) · Zbl 0833.73036
[6] Ghidaglia, J.-M.; Marion, M.; Temam, R., Generalization of the Sobolev-Lieb-Thirring inequalities and applications of attractors, Diff. Integ. Eq., 1, 67-92 (1988)
[7] J. K.Hale,Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, (1988). · Zbl 0642.58013
[8] Hale, J. K.; Raugel, G., Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71, 33-95 (1992) · Zbl 0840.35044
[9] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), New York: Springer, New York · Zbl 0456.35001
[10] Kapitansky, L. V.; Kostin, I. N., Attractors of non-linear evolution equations and their approximations, Leningrad Math. J., 2, 97-117 (1991) · Zbl 0724.35049
[11] Khruslov, E. Ya., Homogenized diffusion model in cracked-porous media, Dokl. Akad. Nauk SSSR, 309, 332-335 (1989)
[12] Khruslov, E. Ya., Homogenized model of strongly inhomogeneous medium with memory, Uspeckhi Math. Nauk, 45, 1, 197-199 (1990)
[13] E.Ya. Khruslov,Homogenized models of composite media, inComposite Media and Homogenized Theory (Eds. G. DalMaso, G. F.Dell’Antonio) Birkhauser (1991), pp. 159-182. · Zbl 0737.73009
[14] E.Ya. Khruslov,Homogenized models of strongly inhomogeneous media, talk on Math. Congress Zurich (1994) (to be published).
[15] Ladyzenskaya, O. A.; Solonnikov, V. A.; Uraltseva, N. N., Linear and quasilinear equations of parabolic type (1968), Providence: Amer. Math. Soc., Providence
[16] Morita, J.; Jimbo, S., Ordinary differential equations on inertial manifolds for reaction-diffusion systems in a singularity perturbed domain with several thin channels, J. Dyn. Diff. Eqs., 4, 65-93 (1992) · Zbl 0760.35026
[17] Temam, R., Infinite dimensional systems, Mechanics and Physics (1988), New York: Springer, New York
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