Averaging of nonautonomous damped wave equations with singularly oscillating external forces

摘要

We consider for rho is an element of [0, 1] and epsilon > 0 small, the nonautonomous weakly damped wave equation With a Singularly oscillating external force partial derivative(2)(t)u - Delta u + gamma partial derivative(t)u = - f(u) + g(0)(t) + epsilon(-rho) g(1)(t/epsilon), together with the averaged equation partial derivative(2)(t)u - Delta u + gamma partial derivative(t)u = -f(u) + g(0)(t). Under suitable assumptions oil the nonlinearity and the external force, we prove the uniform (with respect to epsilon) boundedness of the attractors A(epsilon) in the weak energy space. If rho < 1, we establish the convergence of the attractor A(epsilon) of the first equation to the attractor A(0) ofthe second one, as epsilon -> 0(+). On the other hand, if rho = 1, this convergence may fail. When A(0) is exponential, then the convergence rate of A(epsilon) to A(0) is controlled by M epsilon(eta), for some M >= 0 and some eta = eta(rho) is an element of (0, 1). (c) 2008 Elsevier Masson SAS. All rights reserved.