Fenfen Wang, Rafael de la Llave
Response solutions to quasi-periodically forced systems, even to possibly ill-posed PDEs, with strongly dissipation and any frequency vectors
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ABSTRACT. 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
$arepsilon$ (where $arepsilon$ is the inverse of the coefficients
multiplying the damping) when we consider $arepsilon$ in a domain that does not include the
origin $arepsilon=0$ but has the origin on its boundary. We get that the response solutions depend continuously on
$arepsilon$ when we consider $arepsilon $ tends to $0$. However, in general, they may not be differentiable at $arepsilon=0$. 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 is that it gives results for analytic,
finitely differentiable and low regularity forcing and nonlinearity, respectively. As a matter of fact,
we do not even need that the forcing is continuous. Notably, we obtain
results when the forcing is in $L^2$ space and the nonlinearity is just
Lipschitz as well as in the case that the forcing is in $H^1$ space and the
nonlinearity is $C^{1 + ext{Lip}}$.
In the proof of our results, we reformulate the existence of
response solutions as a fixed point problem in appropriate
spaces of smooth functions. Based on the fixed point problem, we will obtain response solutions as well as some regularity with respect to the singular
perturbation parameter $arepsilon$ except at the origin $arepsilon=0$. More precisely, in the analytic case, we use the contraction mapping principle to get
response
solutions analytic in $arepsilon$ for $arepsilon$ in a complex domain. In the highly differentiable case, to obtain optimal regularity in $arepsilon$, we combine with the classical implicit function theorem. In the low regularity case, such as $H^1$, the contraction argument we use will be somewhat more sophisticated. Particularly, we do not use dynamical
properties of the models, so the method applies even to ill-posed equations and we give some examples.