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The dynamics of fast non-autonomous travelling waves and homogenization. (English) Zbl 1004.35010

Carrive, M. (ed.) et al., Actes des journées “Jeunes numériciens” en l’honneur du 60ème anniversaire du Professeur Roger Temam, Poitiers, France, Mars 9-10, 2000. Poitou-Charentes: Atlantique. 131-142 (2001).
The author considers the boundary value problem \[ a(\partial_t^2u+\Delta_xu) - (\gamma/\varepsilon)\partial_tu - f(u) = g(t);\quad u|_{\partial\Omega}=0;\quad u|_{t=0}=u_0 \] in a semicylinder \(\Omega_{+}=\mathbb R_{+}\times\omega\), where \(\omega\) is a bounded smooth domain in \(\mathbb R^n\), \(u=(u^1,\dots,u^k)\) is an unknown vector function, \(f\), \(g\), and \(u_0\) are given vector functions, \(a\) and \(\gamma\) are given constant \(k\times k\)-matrices such that \(a+a^\ast >0\) and \(\gamma=\gamma^\ast >0\), and \(\varepsilon\) is a small positive parameter (\(\varepsilon \ll 1\)). It is assumed that the function \(f\) satisfies the following inequalities: \(f(v)\cdot v \geqslant -C\), \(f^\prime(v)\geqslant -K\), \(|f(v)|\leqslant C(1+|v|^q)\) with appropriate constants \(C\) and \(K\), an exponent \(q< (n+2)/(n-2)\) and every \(v\in \mathbb R^k\). It is supposed also that \(g\in C_b(\mathbb R, L^2(\omega))\) and \(g\) is almost periodic. The aim of the paper is to study the behaviour of the attractors \(\mathcal A^\varepsilon\) for the problem under consideration as \(\varepsilon \to 0\).
For the entire collection see [Zbl 0978.00050].

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L70 Second-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35B25 Singular perturbations in context of PDEs
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