Attractors of dissipative hyperbolic equations with singularly oscillating external forces (Q881026)

The authors study a uniform attractor \(A^\varepsilon\) for a nonautonomous dissipative wave equation \[ \partial_t^2u+\gamma\partial_tu=\Delta u-f(u)+g^\varepsilon(x,t) \] in a bounded domain \(\Omega\subset{\mathbb R}^n\) under the assumption that the external force \(g^\varepsilon(x,t):=g_0(x,t)+\varepsilon^{-\alpha}g_1(x,t/\varepsilon)\) singularly oscillates in time, where \(\alpha>0\), \(\varepsilon\in(0,1]\). In the space \(E:=H_0^1\times L_2\) this equation admits an absorbing set \(B^\varepsilon\), which can be estimated as \(\| B^\varepsilon\| _E\leq C_1+C_2\varepsilon^{-\alpha}\) and, therefore, can increase without bound in the norm of \(E\) as \(\varepsilon\searrow 0\). Under certain constraints on the function \(g_1\) it is shown that for sufficiently small \(\alpha>0\) the global attractors \(A^\varepsilon\), \(\varepsilon\in(0,1]\), of such a perturbed equation are bounded in \(E\). Along with the original equation, a ``limiting'' wave equation with external force \(g_0(x,t)\) is considered, which also has a global attractor \(A^0\). For the autonomous case \(g_0(x,t)=g_0(x)\) and an exponential global attractor \(A^0\) of the limiting equation, the estimate \(\text{dist}_E(A^0,A^\varepsilon)\leq C\varepsilon^{\eta(\alpha)}\) for the Hausdorff distance is established, where an explicit formula for \(\eta(\alpha)>0\) is given. Additionally, the nonautonomous case is studied, in which \(g_0\) depends on time. Here, it is assumed that sufficient conditions are satisfied for which the ``limiting equation'' has an exponential global attractor. In this case, similar upper bounds for the Hausdorff distance between \(A^0\) and \(A^\varepsilon\) are given.