Wave equation in higher dimensions - periodic solutions

Andrzej Nowakowski,Andrzej Rogowski

doi:10.14232/ejqtde.2018.1.103

Abstract

Wave equation in higher dimensions - periodic solutions

Highlights

The aim of the paper is to look for solutions and their regularity to the following problem xtt(t, y) − ∆x(t, y) + l(t, y, x(t, y)) = 0, t ∈ R, y ∈ (0, π)n, x(t, y) = 0, y ∈ ∂(0, π)n, t ∈ R, (1.1)x(t + T, y) = x(t, y), t ∈ R, y ∈ (0, π)n.The study of time periodic solutions to (1.1), typically with T = 2π, has a long history
As the last step we investigate a certain form of l being a special combination of a finite number of increasing functions to which we apply induction method and use the obtained result for difference of two monotone functions
In this paper we extend nonlinearity to be a finite linear combination of monotone functions

Summary

Introduction

The main limitation of this method is the fact that standard KAM-techniques require the linear frequencies to be well separated (non resonance between the linear frequencies) To overcome such difficulty a new method for proving the existence of small amplitude periodic solutions, based on the Lyapunov–Schmidt reduction, has been developed in [18]. The aim of this paper is to consider the case n ≥ 2 with T being irrational numbers such that α = T/π has not necessary bounded partial quotients in its continued fraction and nonautonomous nonlinearity l. That means first we define a functional of convex type (l is monotone only) and using duality properties of convex analysis we prove existence and regularity of solution to (1.1) as a minimum of the functional on a suitable defined set depending on the type of irrational frequency. From the above we infer that the set of α satisfying T is nonempty, in the following sense: there exists α irrational and r ≥ 2 satisfying T with some constant c > 0 (compare (1.5))!

Main results

FnG and

There exist constants

FG and notice

More general case – proof of the main theorem

Conclusions