Homogenization of attractors of nonlinear evolutionary equations (Q2735870)

The paper is devoted to the homogenization of nonlinear evolutionary equations in domains with ``traps''. The authors consider an initial boundary value problem for a semilinear parabolic equation of the form: NEWLINE\[NEWLINE\frac{\partial u}{\partial t}- L_\varepsilon u+ f(u)= h, \quad t>0, \qquad u^\varepsilon (x,0)= u_0^\varepsilon (x),NEWLINE\]NEWLINE where \(L_\varepsilon\) is a linear second order differential operator with corresponding boundary condition.NEWLINENEWLINEThe application of standard methods makes possible to prove that the dynamical system for every \(\varepsilon> 0\) has a compact global attractor \(A_\varepsilon\) and this attractor has a finite Hausdorff dimension. It is shown that the homogenization of such a problem leads to a system of a semilinear parabolic equation coupled with an ordinary differential equation which possesses a finite-dimensional global attractor \(A\). The authors investigate properties of \(A\) and prove that the global attractors \(A_\varepsilon\) tend to \(A\) in a suitable sense as \(\varepsilon\to 0\).