Quasi-periodic response solutions of nonlinear plate models with nonlocal energy damping

Abstract

Response solutions are quasi-periodic ones with the same frequency as the forcing term. The present work is devoted to constructing response solutions for $d$-dimensional nonlinear plate models with nonlocal energy damping, which are closely related to damping phenomena in flight structures. For such models, the main characteristic is that the dissipation rate depends on the energy strength. By considering a small parameter $\epsilon$ in the domain excluding the origin and imposing a small quasi-periodic forcing with a Diophantine frequency vector, we demonstrate the persistence of the corresponding response solution. We provide an alternative approach to the contraction mapping principle (cf. [7, 33]) through a combination of reduction together with the Nash--Moser iteration technique. The reason behind this approach lies in the derivative losses caused by the nonlocal nonlinearity.