Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation
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Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation Livia Corsi2 · Riccardo Montalto1
· Michela Procesi2
Received: 31 May 2020 / Accepted: 10 October 2020 © The Author(s) 2020
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 T∞ . Keywords Almost-periodic solutions for PDEs · Nash–Moser-KAM theory · Small divisor problems · KdV Mathematics Subject Classification 37K55 · 58C15 · 35Q53 · 35B15
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Functional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Zero-th Step . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The n + 1-th Step . . . . . . . . . . . . . . . . . . . . . . . . 4 Proof of Proposition 3.6 . . . . . . . . . . . . . . . . . . . . . . . 4.1 Elimination of the x-Dependence from the Highest Order Term 4.2 Elimination of the ϕ-Dependence from the Highest Order Term 4.3 Time Dependent Traslation of the Space Variable . . . . . . . 4.4 Conclusion of the Proof . . . . . . . . . . . . . . . . . . . . . 5 Proof of Proposition 3.8 . . . . . . . . . . . . . . . . . . . . . . .
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Riccardo Montalto [email protected] Livia Corsi [email protected] Michela Procesi [email protected]
1
Università degli Studi di Milano, Milan, Italy
2
Università di Roma Tre, Rome, Italy
123
Journal of Dynamics and Differential Equations 5.1 Reduction of the First Order Term 5.2 Reducibility . . . . . . . . . . . . 5.3 Variations . . . . . . . . . . . . . 5.4 Conclusion of the Proof . . . . . . A Technical Lemmata . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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1 Introduction In this paper we study response solutions for almost-periodically forced quasilinear PDEs close to an elliptic fixed poi
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