The dynamics of fast non-autonomous travelling waves and homogenization (Q2782331)

The author considers the boundary value problem NEWLINE\[NEWLINE a(\partial_t^2u+\Delta_xu) - (\gamma/\varepsilon)\partial_tu - f(u) = g(t);\quad u|_{\partial\Omega}=0;\quad u|_{t=0}=u_0 NEWLINE\]NEWLINE 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 \(\varepsilon \to 0\).NEWLINENEWLINEFor the entire collection see [Zbl 0978.00050].