Abstract
We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on mathbb {T}^infty .
Highlights
In this paper we study response solutions for almost-periodically forced quasilinear PDEs close to an elliptic fixed point
The problem of response solutions for PDEs has been widely studied in many contexts, starting from the papers [24,25], where the Author considers a periodically forced PDE with dissipation
In the present paper we study the existence of almost-periodic response solutions, for a quasi-linear PDE on T
Summary
In this paper we study response solutions for almost-periodically forced quasilinear PDEs close to an elliptic fixed point. All the results mentioned above concern semi-linear PDEs and the forcing is quasi-periodic. Concern semi-linear PDEs, with no derivative in the nonlinearity They require a very strong analyticity condition on the forcing term. In the case of a quasi-linear PDE this amounts to study an unbounded non-constant coefficients operator. To deal with this problem, at each step we introduce a change of variables Tn which diagonalizes the highest order terms of the linearized operator. In order to prove the invertibility of the linearized operator after the change of variables Tn is applied, one needs to perform a reducibility scheme: this is done in Sect. For a more detailed description of the technical aspects see Remark 3.2
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