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Response solutions to quasi-periodically forced systems, even to possibly ill-posed PDEs, with strong dissipation and any frequency vectors by Wang, Fenfen; de la Llave, Rafael is a scholarly article available to read on EtoBox.

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We consider several models (including both multidimensional ordinary differential equations (ODEs) and partial differential equations (PDEs), possibly ill-posed), subject to very strong damping and quasi-periodic external forcing. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under some regularity assumptions on the nonlinearity and forcing, without any arithmetic condition on the forcing frequency $\omega$, we show that the response solutions indeed exist. Moreover, the solutions we obtained possess optimal regularity in $\varepsilon$ (where $\varepsilon$ is the inverse of the coefficients multiplying the damping) when we consider $\varepsilon$ in a domain that does not include the origin $\varepsilon=0$ but has the origin on its boundary. We get that the response solutions depend continuously on $\varepsilon$ when we consider $\varepsilon $ tends to $0$. However, in general, they may not be differentiable at $\varepsilon=0$. In this paper, we allow multidimensional systems and we do not require that the unperturbed equations under consideration are Hamiltonian. One advantage of the method in the present paper

Author

Wang, Fenfen; de la Llave, Rafael

Published

2019

Language

EN