Attractors of dissipative hyperbolic equations with singularly oscillating external forces

摘要

We study a uniform attractor A for a dissipative wave equation in a bounded domain Ω □□n under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g 0(x, t)+ ε?α g 1 (x, t/ε), x ∈ Ω, t ∈ □, where α > 0, 0 < ε ≤ 1. In E = H 0 1 × L 2, this equation has an absorbing set B ε estimated as ‖B ε‖ E ≤C 1+C 2ε?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g 1(x, z), x ∈ Ω, z ∈ □, we prove that, for 0 < α ≤ α 0, the global attractors of such an equation are bounded in E, i.e.,A~ε||E≤C_3 , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g 0(x, t) that also has a global attractor . For the case in which g 0(x, t) = g 0(x) and the global attractor of the limiting equation is exponential, it is established that, for 0 < α ≤ α 0, the Hausdorff distance satisfies the estimate dist _E(A~ε,A~o)≤Cεn(a), where η(α) > 0. For η(α) and α 0, explicit formulas are given. We also study the nonautonomous case in which g 0 = g 0(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors A_ε from A~o , similar to those given above.