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

TL;DR

The paper studies a d-dimensional beam/plate equation with nonlocal energy damping and quasi-periodic forcing, given by $u_{tt}+\lambda\Delta^2u-\mu\Delta u+\epsilon u_t+\alpha(\lambda\int_{\mathbb{T}^d}|\Delta u|^2dx+\int_{\mathbb{T}^d}|u_t|^2dx)u_t=\epsilon^2 g(\omega t,x)$. After a phase shift $\varphi=\omega t$ and a rescaling $u\mapsto \epsilon^p u$, the problem is recast as a nonlinear equation with a quasi-periodic forcing term and a nonlocal nonlinearity $F(u)$, and a main result asserts the existence of a zero-mean quasi-periodic solution in $H^{\rho_0/2,s}_0$ for small $\epsilon$ under a Diophantine non-resonance condition on $\omega$. The authors develop a reducibility scheme to conjugate the linearized operator to a constant-coefficient form $\mathcal{D}$ plus a small remainder $\tilde{\mathcal{R}}$, and obtain sharp inverse estimates for $\mathcal{L}^{-1}$ via a diagonalization in Fourier space and a Neumann-series argument. Combining this linear analysis with a Nash–Moser iteration on finite-dimensional spaces, they prove existence of a quasi-periodic, zero-mean solution and outline extensions to singular perturbation PDEs with nonlocal damping, contributing a robust framework for response solutions in nonlinear PDEs with energy-dependent 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 $ε$ 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.

Theorems & Definitions (28)

• Remark 2.1
• Theorem 2.2
• Proposition 3.1
• Lemma 3.2
• proof
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5